var bibbase_data = {"data":"<img src=\"//bibbase.org/img/ajax-loader.gif\" id=\"spinner\"\n  style=\"display: none;\" alt=\"Loading..\" />\n\n<div id=\"bibbase\">\n\n  <script type=\"text/javascript\">\n    var bibbase = {\n      params: {\"bib\":\"https://drive.google.com/uc?export=download&id=1NEXsxwRAx2CWt43v0y0jyZdP1LdOS_xF\",\"jsonp\":\"1\",\"authorFirst\":\"1\",\"sort\":\"-year\",\"theme\":\"side\",\"filter\":\"keywords:ACSV\",\"host\":\"bibbase.org\"},\n      data: [{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Greenwood\"],\"firstnames\":[\"Torin\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Melczer\"],\"firstnames\":[\"Stephen\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Ruza\"],\"firstnames\":[\"Tiadora\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Wilson\"],\"firstnames\":[\"Mark\",\"C.\"],\"suffixes\":[]}],\"journal\":\"Séminaire Lotharingien de Combinatoire\",\"abstract\":\"We present a strategy for computing asymptotics of coefficients of $d$-variate algebraic generating functions. Using known constructions, we embed the coefficient array into an array represented by a rational generating functions in $d+1$ variables, and then apply ACSV theory to analyse the latter. This method allows us to give systematic results in the multivariate case, seems more promising than trying to derive analogs of the rational ACSV theory for algebraic GFs, and gives the prospect of further improvements as embedding methods are studied in more detail.\",\"title\":\"Asymptotics of coefficients of algebraic series via embedding into rational series (extended abstract)\",\"year\":\"2022\",\"pages\":\"12pp\",\"volume\":\"86B\",\"keywords\":\"ACSV applications, ACSV theory\",\"url_paper\":\"https://www.emis.de/journals/SLC/wpapers/FPSAC2022/30.pdf\",\"url_slides\":\"\",\"bibtex\":\"@Article{GMRW2022,\\nauthor = {Greenwood, Torin and Melczer, Stephen and Ruza, Tiadora and Wilson, Mark C.},\\njournal = {S\\\\'{e}minaire Lotharingien de Combinatoire},\\nabstract = {We present a strategy for computing asymptotics of coefficients of\\n$d$-variate algebraic generating functions. Using known constructions,\\nwe embed the coefficient array into an array represented by a rational\\ngenerating functions in $d+1$ variables, and then apply ACSV theory to\\nanalyse the latter. This method allows us to give systematic results in\\nthe multivariate case, seems more promising than trying to derive\\nanalogs of the rational ACSV theory for algebraic GFs, and gives the\\nprospect of further improvements as embedding methods are studied in\\nmore detail.},\\ntitle = {Asymptotics of coefficients of algebraic series via embedding into rational series (extended abstract)},\\nyear = {2022},\\npages = {12pp},\\nvolume = {86B}, \\nkeywords={ACSV applications, ACSV theory},\\nurl_paper = {https://www.emis.de/journals/SLC/wpapers/FPSAC2022/30.pdf},\\nurl_slides = {},\\n}\\n\\n\",\"author_short\":[\"Greenwood, T.\",\"Melczer, S.\",\"Ruza, T.\",\"Wilson, M. C.\"],\"key\":\"GMRW2022\",\"id\":\"GMRW2022\",\"bibbaseid\":\"greenwood-melczer-ruza-wilson-asymptoticsofcoefficientsofalgebraicseriesviaembeddingintorationalseriesextendedabstract-2022\",\"role\":\"author\",\"urls\":{\" paper\":\"https://www.emis.de/journals/SLC/wpapers/FPSAC2022/30.pdf\",\" slides\":\"https://drive.google.com/uc?export=download&id=1NEXsxwRAx2CWt43v0y0jyZdP1LdOS_xF\"},\"keyword\":[\"ACSV applications\",\"ACSV theory\"],\"metadata\":{\"authorlinks\":{\"wilson, m\":\"https://markcwilson.site/Research/ACSV/\"}},\"downloads\":4,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"title\":\"Quiver asymptotics: free chiral ring\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Ramgoolam\"],\"firstnames\":[\"Sanjaye\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Wilson\"],\"firstnames\":[\"Mark\",\"C\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Zahabi\"],\"firstnames\":[\"Ali\"],\"suffixes\":[]}],\"journal\":\"Journal of Physics A: Mathematical and Theoretical\",\"volume\":\"53\",\"number\":\"10\",\"pages\":\"105401\",\"year\":\"2020\",\"publisher\":\"IOP Publishing\",\"keywords\":\"ACSV applications\",\"url_paper\":\"https://iopscience.iop.org/article/10.1088/1751-8121/ab6fc6/pdf\",\"abstract\":\"The large N generating functions for the counting of chiral operators in $\\\\mathcal{N} = 1$, four-dimensional quiver gauge theories have previously been obtained in terms of the weighted adjacency matrix of the quiver diagram. We introduce the methods of multi-variate asymptotic analysis to study this counting in the limit of large charges. We describe a Hagedorn phase transition associated with this asymptotics, which refines and generalizes known results on the 2-matrix harmonic oscillator. Explicit results are obtained for two infinite classes of quiver theories, namely the generalized clover quivers and affine $ℂ^3 ∖ Â_n$ orbifold quivers.\",\"bibtex\":\"@article{ramgoolam2020quiver,\\n  title={Quiver asymptotics: free chiral ring},\\n  author={Ramgoolam, Sanjaye and Wilson, Mark C and Zahabi, Ali},\\n  journal={Journal of Physics A: Mathematical and Theoretical},\\n  volume={53},\\n  number={10},\\n  pages={105401},\\n  year={2020},\\n  publisher={IOP Publishing},\\n  keywords={ACSV applications},\\n  url_Paper={https://iopscience.iop.org/article/10.1088/1751-8121/ab6fc6/pdf},\\n  abstract={The large N generating functions for the counting of chiral operators in\\n$\\\\mathcal{N} = 1$, four-dimensional quiver gauge theories have\\npreviously been obtained in terms of the weighted adjacency matrix of\\nthe quiver diagram. We introduce the methods of multi-variate asymptotic\\nanalysis to study this counting in the limit of large charges. We\\ndescribe a Hagedorn phase transition associated with this asymptotics,\\nwhich refines and generalizes known results on the 2-matrix harmonic\\noscillator. Explicit results are obtained for two infinite classes of\\nquiver theories, namely the generalized clover quivers and affine\\n$\\\\mathbb{C}^3 \\\\setminus \\\\hat{A}_n$ orbifold quivers.}\\n}\\n\\n\\n\\n\",\"author_short\":[\"Ramgoolam, S.\",\"Wilson, M. C\",\"Zahabi, A.\"],\"key\":\"ramgoolam2020quiver\",\"id\":\"ramgoolam2020quiver\",\"bibbaseid\":\"ramgoolam-wilson-zahabi-quiverasymptoticsfreechiralring-2020\",\"role\":\"author\",\"urls\":{\" paper\":\"https://iopscience.iop.org/article/10.1088/1751-8121/ab6fc6/pdf\"},\"keyword\":[\"ACSV applications\"],\"metadata\":{\"authorlinks\":{\"wilson, m\":\"https://markcwilson.site/Research/ACSV/\"}},\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Melczer\"],\"firstnames\":[\"Stephen\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Wilson\"],\"firstnames\":[\"Mark\",\"C.\"],\"suffixes\":[]}],\"title\":\"Higher dimensional lattice walks: Connecting combinatorial and analytic behavior\",\"number\":\"4\",\"pages\":\"2140-2174\",\"volume\":\"33\",\"abstract\":\"We consider the enumeration of walks on the non-negative lattice $ℕ^d$, with steps defined by a set $\\\\mathcal{S} ⊂ \\\\{ -1, 0, 1 \\\\}^d ∖ \\\\{\\\\mathbf{0}\\\\}$. Previous work in this area has established asymptotics for the number of walks in certain families of models by applying the techniques of analytic combinatorics in several variables (ACSV), where one encodes the generating function of a lattice path model as the diagonal of a multivariate rational function. Melczer and Mishna obtained asymptotics when the set of steps $\\\\mathcal{S}$ is symmetric over every axis; in this setting one can always apply the methods of ACSV to a multivariate rational function whose set of singularities is a smooth manifold (the simplest case). Here we go further, providing asymptotics for models with generating functions that must be encoded by multivariate rational functions having non-smooth singular sets. In the process, our analysis connects past work to deeper structural results in the theory of analytic combinatorics in several variables. One application is a closed form for asymptotics of models defined by step sets that are symmetric over all but one axis. As a special case, we apply our results when $d=2$ to give a rigorous proof of asymptotics conjectured by Bostan and Kauers; asymptotics for walks returning to boundary axes and the origin are also given.\",\"journal\":\"SIAM Journal on Discrete Mathematics\",\"keywords\":\"ACSV applications\",\"publisher\":\"Society for Industrial and Applied Mathematics\",\"url_paper\":\"https://arxiv.org/abs/1810.06170\",\"url_slides\":\"https://www.youtube.com/watch?v=o5SOHJbGO4g\",\"year\":\"2019\",\"bibtex\":\"@Article{melczer2019higher,\\n  author     = {Melczer, Stephen and Wilson, Mark C.},\\n  title      = {Higher dimensional lattice walks: Connecting combinatorial and analytic behavior},\\n  number     = {4},\\n  pages      = {2140-2174},\\n  volume     = {33},\\n  abstract   = {We consider the enumeration of walks on the non-negative lattice\\n$\\\\mathbb{N}^d$, with steps defined by a set $\\\\mathcal{S} \\\\subset \\\\{ -1,\\n0, 1 \\\\}^d \\\\setminus \\\\{\\\\mathbf{0}\\\\}$. Previous work in this area has\\nestablished asymptotics for the number of walks in certain families of\\nmodels by applying the techniques of analytic combinatorics in several\\nvariables (ACSV), where one encodes the generating function of a lattice\\npath model as the diagonal of a multivariate rational function. Melczer\\nand Mishna obtained asymptotics when the set of steps $\\\\mathcal{S}$ is\\nsymmetric over every axis; in this setting one can always apply the\\nmethods of ACSV to a multivariate rational function whose  set of\\nsingularities is a smooth manifold (the simplest case). Here we go\\nfurther, providing asymptotics for models with generating functions that\\nmust be encoded by multivariate rational functions having non-smooth\\nsingular sets.  In the process, our analysis connects past work to\\ndeeper structural results in the theory of analytic combinatorics in\\nseveral variables.  One application is a closed form for asymptotics of\\nmodels defined by step sets that are symmetric over all but one axis. As\\na special case, we apply our results when $d=2$ to give a rigorous proof\\nof asymptotics conjectured by Bostan and Kauers; asymptotics for walks\\nreturning to boundary axes and the origin are also given.},\\n  journal    = {SIAM Journal on Discrete Mathematics},\\n  keywords   = {ACSV applications},\\n  publisher  = {Society for Industrial and Applied Mathematics},\\n  url_paper  = {https://arxiv.org/abs/1810.06170},\\n  url_slides = {https://www.youtube.com/watch?v=o5SOHJbGO4g},\\n  year       = {2019},\\n}\\n\\n\",\"author_short\":[\"Melczer, S.\",\"Wilson, M. C.\"],\"key\":\"melczer2019higher\",\"id\":\"melczer2019higher\",\"bibbaseid\":\"melczer-wilson-higherdimensionallatticewalksconnectingcombinatorialandanalyticbehavior-2019\",\"role\":\"author\",\"urls\":{\" paper\":\"https://arxiv.org/abs/1810.06170\",\" slides\":\"https://www.youtube.com/watch?v=o5SOHJbGO4g\"},\"keyword\":[\"ACSV applications\"],\"metadata\":{\"authorlinks\":{\"wilson, m\":\"https://markcwilson.site/Research/ACSV/\"}},\"html\":\"\"},{\"bibtype\":\"inproceedings\",\"type\":\"inproceedings\",\"author\":[{\"firstnames\":[\"Stephen\"],\"propositions\":[],\"lastnames\":[\"Melczer\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Wilson\"],\"firstnames\":[\"Mark\",\"C.\"],\"suffixes\":[]}],\"title\":\"Asymptotics of lattice walks via analytic combinatorics in several variables\",\"booktitle\":\"2016 Conference on Formal Power Series and Algebraic Combinatorics, FPSAC2016\",\"series\":\"Discrete Math. Theor. Comput. Sci. Proc., AH\",\"pages\":\"863-874\",\"publisher\":\"Assoc. Discrete Math. Theor. Comput. Sci., Nancy\",\"year\":\"2016\",\"keywords\":\"ACSV applications\",\"url_paper\":\"https://dmtcs.episciences.org/6390/pdf\",\"abstract\":\"We consider the enumeration of walks on the two dimensional non-negative integer lattice with short steps. Up to isomorphism there are 79 unique two dimensional models to consider, and previous work in this area has used the kernel method, along with a rigorous computer algebra approach, to show that 23 of the 79 models admit D-finite generating functions. In 2009, Bostan and Kauers used Padé-Hermite approximants to guess differential equations which these 23 generating functions satisfy, in the process guessing asymptotics of their coefficient sequences. In this article we provide, for the first time, a complete rigorous verification of these guesses. Our technique is to use the kernel method to express 19 of the 23 generating functions as diagonals of tri-variate rational functions and apply the methods of analytic combinatorics in several variables (the remaining 4 models have algebraic generating functions and can thus be handled by univariate techniques). This approach also shows the link between combinatorial properties of the models and features of its asymptotics such as asymptotic and polynomial growth factors. In addition, we give expressions for the number of walks returning to the x-axis, the y-axis, and the origin, proving recently conjectured asymptotics of Bostan, Chyzak, van Hoeij, Kauers, and Pech.\",\"bibtex\":\"@inproceedings {MeWi2016, \\nAUTHOR = {Stephen Melczer and Wilson, Mark C.},\\n     TITLE = {Asymptotics of lattice walks via analytic combinatorics in several variables},\\n BOOKTITLE = {2016 {C}onference on {F}ormal {P}ower {S}eries and {A}lgebraic {C}ombinatorics, FPSAC2016},\\n    SERIES = {Discrete Math. Theor. Comput. Sci. Proc., AH},\\n     PAGES = {863-874},\\n PUBLISHER = {Assoc. Discrete Math. Theor. Comput. Sci., Nancy},\\n      YEAR = {2016},\\n  keywords={ACSV applications},\\n  url_Paper={https://dmtcs.episciences.org/6390/pdf},\\n  abstract={We consider the enumeration of walks on the two dimensional non-negative\\ninteger lattice with short steps. Up to isomorphism there are 79 unique\\ntwo dimensional models to consider, and previous work in this area has\\nused the kernel method, along with a rigorous computer algebra approach,\\nto show that 23 of the 79 models admit D-finite generating functions. In\\n2009, Bostan and Kauers used Pad\\\\'{e}-Hermite approximants to guess\\ndifferential equations which these 23 generating functions satisfy, in\\nthe process guessing asymptotics of their coefficient sequences. In this\\narticle we provide, for the first time, a complete rigorous verification\\nof these guesses. Our technique is to use the kernel method to express\\n19 of the 23 generating functions as diagonals of tri-variate rational\\nfunctions and apply the methods of analytic combinatorics in several\\nvariables (the remaining 4 models have algebraic generating functions\\nand can thus be handled by univariate techniques). This approach also\\nshows the link between combinatorial properties of the models and\\nfeatures of its asymptotics such as asymptotic and polynomial growth\\nfactors. In addition, we give expressions for the number of walks\\nreturning to the x-axis, the y-axis, and the origin, proving recently\\nconjectured asymptotics of Bostan, Chyzak, van Hoeij, Kauers, and Pech.}\\n  \\n}\\n\\n\\n\",\"author_short\":[\"Melczer, S.\",\"Wilson, M. C.\"],\"key\":\"MeWi2016\",\"id\":\"MeWi2016\",\"bibbaseid\":\"melczer-wilson-asymptoticsoflatticewalksviaanalyticcombinatoricsinseveralvariables-2016\",\"role\":\"author\",\"urls\":{\" paper\":\"https://dmtcs.episciences.org/6390/pdf\"},\"keyword\":[\"ACSV applications\"],\"metadata\":{\"authorlinks\":{\"wilson, m\":\"https://markcwilson.site/Research/ACSV/\"}},\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"title\":\"Diagonal asymptotics for products of combinatorial classes\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Wilson\"],\"firstnames\":[\"Mark\",\"C.\"],\"suffixes\":[]}],\"journal\":\"Combinatorics, Probability and Computing\",\"volume\":\"24\",\"number\":\"1\",\"pages\":\"354-372\",\"year\":\"2015\",\"publisher\":\"Cambridge University Press\",\"keywords\":\"ACSV theory\",\"url_paper\":\"https://www.cambridge.org/core/journals/combinatorics-probability-and-computing/article/abs/diagonal-asymptotics-for-products-of-combinatorial-classes/DE65A4AA078A6161905DD6CEFBEDC85E\",\"abstract\":\"We generalize and improve recent results by Bóna and Knopfmacher and by Banderier and Hitczenko concerning the joint distribution of the sum and number of parts in tuples of restricted compositions. Specifically, we generalize the problem to general combinatorial classes and relax the requirement that the sizes of the compositions be equal. We extend the main explicit results to enumeration problems whose counting sequences are Riordan arrays. In this framework, we give an alternative method for computing asymptotics in the supercritical case, which avoids explicit diagonal extraction and seems likely to be computationally more efficient.\",\"bibtex\":\"@article{wilson2015diagonal,\\n  title={Diagonal asymptotics for products of combinatorial classes},\\n  author={Wilson, Mark C.},\\n  journal={Combinatorics, Probability and Computing},\\n  volume={24},\\n  number={1},\\n  pages={354-372},\\n  year={2015},\\n  publisher={Cambridge University Press},\\n  keywords={ACSV theory},\\n  url_Paper={https://www.cambridge.org/core/journals/combinatorics-probability-and-computing/article/abs/diagonal-asymptotics-for-products-of-combinatorial-classes/DE65A4AA078A6161905DD6CEFBEDC85E},\\n  abstract={We generalize and improve recent results by B\\\\'{o}na and Knopfmacher and\\nby Banderier and Hitczenko concerning the joint distribution of the sum\\nand number of parts in tuples of restricted compositions. Specifically,\\nwe generalize the problem to general combinatorial classes and relax the\\nrequirement that the sizes of the compositions be equal. We extend the\\nmain explicit results to enumeration problems whose counting sequences\\nare Riordan arrays.  In this framework, we give an alternative method\\nfor computing asymptotics in the supercritical case, which avoids\\nexplicit diagonal extraction and seems likely to be computationally more\\nefficient.}\\n}\\n\\n\",\"author_short\":[\"Wilson, M. C.\"],\"key\":\"wilson2015diagonal\",\"id\":\"wilson2015diagonal\",\"bibbaseid\":\"wilson-diagonalasymptoticsforproductsofcombinatorialclasses-2015\",\"role\":\"author\",\"urls\":{\" paper\":\"https://www.cambridge.org/core/journals/combinatorics-probability-and-computing/article/abs/diagonal-asymptotics-for-products-of-combinatorial-classes/DE65A4AA078A6161905DD6CEFBEDC85E\"},\"keyword\":[\"ACSV theory\"],\"metadata\":{\"authorlinks\":{\"wilson, m\":\"https://markcwilson.site/Research/ACSV/\"}},\"html\":\"\"},{\"bibtype\":\"book\",\"type\":\"book\",\"title\":\"Analytic combinatorics in several variables\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Pemantle\"],\"firstnames\":[\"Robin\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Wilson\"],\"firstnames\":[\"Mark\",\"C.\"],\"suffixes\":[]}],\"year\":\"2013\",\"publisher\":\"Cambridge University Press\",\"keywords\":\"ACSV theory\",\"url_paper\":\"\",\"abstract\":\"\",\"bibtex\":\"@book{pemantle2013analytic,\\n  title={Analytic combinatorics in several variables},\\n  author={Pemantle, Robin and Wilson, Mark C.},\\n  year={2013},\\n  publisher={Cambridge University Press},\\n  keywords={ACSV theory},\\n  url_Paper={},\\n  abstract={}\\n}\\n\\n\",\"author_short\":[\"Pemantle, R.\",\"Wilson, M. C.\"],\"key\":\"pemantle2013analytic\",\"id\":\"pemantle2013analytic\",\"bibbaseid\":\"pemantle-wilson-analyticcombinatoricsinseveralvariables-2013\",\"role\":\"author\",\"urls\":{\" paper\":\"https://drive.google.com/uc?export=download&id=1NEXsxwRAx2CWt43v0y0jyZdP1LdOS_xF\"},\"keyword\":[\"ACSV theory\"],\"metadata\":{\"authorlinks\":{\"wilson, m\":\"https://markcwilson.site/Research/ACSV/\"}},\"downloads\":2,\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"title\":\"A new approach to asymptotics of Maclaurin coefficients of algebraic functions\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Raichev\"],\"firstnames\":[\"Alexander\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Wilson\"],\"firstnames\":[\"Mark\",\"C.\"],\"suffixes\":[]}],\"journal\":\"arXiv preprint arXiv:1202.3826\",\"year\":\"2012\",\"keywords\":\"ACSV theory\",\"url_paper\":\"https://arxiv.org/pdf/1202.3826.pdf\",\"abstract\":\"We propose a general method for deriving asymptotics of the Maclaurin series coefficients of algebraic functions that is based on a procedure of K. V. Safonov and multivariate singularity analysis. We test the feasibility of this this approach by experimenting on several examples.\",\"bibtex\":\"@article{raichev2012new,\\n  title={A new approach to asymptotics of Maclaurin coefficients of algebraic functions},\\n  author={Raichev, Alexander and Wilson, Mark C.},\\n  journal={arXiv preprint arXiv:1202.3826},\\n  year={2012},\\n  keywords={ACSV theory},\\n  url_Paper={https://arxiv.org/pdf/1202.3826.pdf},\\n  abstract={We propose a general method for deriving asymptotics of the Maclaurin\\nseries coefficients of algebraic functions that is based on a procedure of K. V. Safonov\\nand multivariate singularity analysis. We test the feasibility of this this approach by\\nexperimenting on several examples.}\\n}\\n\\n\",\"author_short\":[\"Raichev, A.\",\"Wilson, M. C.\"],\"key\":\"raichev2012new\",\"id\":\"raichev2012new\",\"bibbaseid\":\"raichev-wilson-anewapproachtoasymptoticsofmaclaurincoefficientsofalgebraicfunctions-2012\",\"role\":\"author\",\"urls\":{\" paper\":\"https://arxiv.org/pdf/1202.3826.pdf\"},\"keyword\":[\"ACSV theory\"],\"metadata\":{\"authorlinks\":{\"wilson, m\":\"https://markcwilson.site/Research/ACSV/\"}},\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"title\":\"Asymptotics of coefficients of multivariate generating functions: improvements for multiple points\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Raichev\"],\"firstnames\":[\"Alexander\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Wilson\"],\"firstnames\":[\"Mark\",\"C.\"],\"suffixes\":[]}],\"journal\":\"arXiv preprint arXiv:1009.5715\",\"year\":\"2010\",\"keywords\":\"ACSV theory\",\"url_paper\":\"\",\"abstract\":\"Let $F(x)= ∑_{ν∈ℕ^d} F_ν x^ν$ be a multivariate power series with complex coefficients that converges in a neighborhood of the origin. Assume $F=G/H$ for some functions $G$ and $H$ holomorphic in a neighborhood of the origin. We derive asymptotics for the coefficients $F_{rα}$ as $r \\\\to ∞$ with $rα ∈ ℕ^d$ for $α$ in a permissible subset of $d$-tuples of positive reals. More specifically, we give an algorithm for computing arbitrary terms of the asymptotic expansion for $F_{rα}$ when the asymptotics are controlled by a transverse multiple point of the analytic variety $H = 0$. This improves upon earlier work by R. Pemantle and M. C. Wilson. We have implemented our algorithm in Sage and apply it to obtain accurate numerical results for several rational combinatorial generating functions.\",\"bibtex\":\"@article{raichev2010asymptotics,\\n  title={Asymptotics of coefficients of multivariate generating functions: improvements for multiple points},\\n  author={Raichev, Alexander and Wilson, Mark C.},\\n  journal={arXiv preprint arXiv:1009.5715},\\n  year={2010},\\n  keywords={ACSV theory},\\n  url_Paper={},\\n  abstract={Let $F(x)= \\\\sum_{\\\\nu\\\\in\\\\mathbb{N}^d} F_\\\\nu x^\\\\nu$ be a multivariate\\npower series with complex coefficients that converges in a neighborhood\\nof the origin. Assume $F=G/H$ for some functions $G$ and $H$ holomorphic\\nin a neighborhood of the origin. We derive asymptotics for the\\ncoefficients $F_{r\\\\alpha}$ as $r \\\\to \\\\infty$ with $r\\\\alpha \\\\in\\n\\\\mathbb{N}^d$ for $\\\\alpha$ in a permissible subset of $d$-tuples of\\npositive reals. More specifically, we give an algorithm for computing\\narbitrary terms of the asymptotic expansion for $F_{r\\\\alpha}$ when the\\nasymptotics are controlled by a transverse multiple point of the\\nanalytic variety $H = 0$. This improves upon earlier work  by R.\\nPemantle and M. C. Wilson. We have implemented our algorithm in Sage and\\napply it to obtain accurate numerical results for several rational\\ncombinatorial generating functions.}\\n}\\n\\n\",\"author_short\":[\"Raichev, A.\",\"Wilson, M. C.\"],\"key\":\"raichev2010asymptotics\",\"id\":\"raichev2010asymptotics\",\"bibbaseid\":\"raichev-wilson-asymptoticsofcoefficientsofmultivariategeneratingfunctionsimprovementsformultiplepoints-2010\",\"role\":\"author\",\"urls\":{\" paper\":\"https://drive.google.com/uc?export=download&id=1NEXsxwRAx2CWt43v0y0jyZdP1LdOS_xF\"},\"keyword\":[\"ACSV theory\"],\"metadata\":{\"authorlinks\":{\"wilson, m\":\"https://markcwilson.site/Research/ACSV/\"}},\"html\":\"\"},{\"bibtype\":\"incollection\",\"type\":\"incollection\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Pemantle\"],\"firstnames\":[\"Robin\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Wilson\"],\"firstnames\":[\"Mark\",\"C.\"],\"suffixes\":[]}],\"title\":\"Asymptotic expansions of oscillatory integrals with complex phase\",\"booktitle\":\"Algorithmic probability and combinatorics\",\"series\":\"Contemp. Math.\",\"volume\":\"520\",\"pages\":\"221-240\",\"publisher\":\"Amer. Math. Soc., Providence, RI\",\"year\":\"2010\",\"keywords\":\"ACSV theory\",\"url_paper\":\"https://arxiv.org/pdf/0903.3585.pdf\",\"abstract\":\"We consider saddle point integrals in $d$ variables whose phase function is neither real nor purely imaginary. Results analogous to those for Laplace (real phase) and Fourier (imaginary phase) integrals hold whenever the phase function is analytic and nondegenerate. These results generalize what is well known for integrals of Laplace and Fourier type. The method is via contour shifting in complex $d$-space. This work is motivated by applications to asymptotic enumeration.\",\"bibtex\":\"@incollection {PeWi2010,\\n    AUTHOR = {Pemantle, Robin and Wilson, Mark C.},\\n     TITLE = {Asymptotic expansions of oscillatory integrals with complex\\n              phase},\\n BOOKTITLE = {Algorithmic probability and combinatorics},\\n    SERIES = {Contemp. Math.},\\n    VOLUME = {520},\\n     PAGES = {221-240},\\n PUBLISHER = {Amer. Math. Soc., Providence, RI},\\n      YEAR = {2010},\\n       keywords={ACSV theory},\\n  url_Paper={https://arxiv.org/pdf/0903.3585.pdf},\\n  abstract={We consider saddle point integrals in $d$ variables whose phase function\\nis neither real nor purely imaginary. Results analogous to those for\\nLaplace (real phase) and Fourier (imaginary phase) integrals hold\\nwhenever the phase function is analytic and nondegenerate. These results\\ngeneralize what is well known for integrals of Laplace and Fourier type.\\nThe method is via contour shifting in complex $d$-space. This work is\\nmotivated by applications to asymptotic enumeration.}\\n}\\n\\n\",\"author_short\":[\"Pemantle, R.\",\"Wilson, M. C.\"],\"key\":\"PeWi2010\",\"id\":\"PeWi2010\",\"bibbaseid\":\"pemantle-wilson-asymptoticexpansionsofoscillatoryintegralswithcomplexphase-2010\",\"role\":\"author\",\"urls\":{\" paper\":\"https://arxiv.org/pdf/0903.3585.pdf\"},\"keyword\":[\"ACSV theory\"],\"metadata\":{\"authorlinks\":{\"wilson, m\":\"https://markcwilson.site/Research/ACSV/\"}},\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"title\":\"Twenty combinatorial examples of asymptotics derived from multivariate generating functions\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Pemantle\"],\"firstnames\":[\"Robin\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Wilson\"],\"firstnames\":[\"Mark\",\"C.\"],\"suffixes\":[]}],\"journal\":\"SIAM Review\",\"volume\":\"50\",\"number\":\"2\",\"pages\":\"199-272\",\"year\":\"2008\",\"publisher\":\"Society for Industrial and Applied Mathematics\",\"keywords\":\"ACSV theory\",\"url_paper\":\"https://epubs.siam.org/doi/epdf/10.1137/050643866\",\"abstract\":\"Let $F$ be a power series in at least two variables that defines a meromorphic function in a neighbourhood of the origin; for example, $F$ may be a rational multivariate generating function. We discuss recent results that allow the effective computation of asymptotic expansions for the coefficients of $F$, uniform in certain explicitly defined cones of directions. The purpose of this article is to illustrate the use of these techniques on a variety of problems of combinatorial interest. The first part reviews the Morse-theoretic underpinnings of these techniques, and then summarizes the necessary results so that only elementary analyses are needed to check hypotheses and carry out computations. The remainder focuses on combinatorial applications. Specific examples deal with enumeration of words with forbidden substrings, edges and cycles in graphs, polyominoes, descents and solutions to integer equations. After the individual examples, we discuss three broad classes of examples, namely functions derived via the transfer matrix method, those derived via the kernel method, and those derived via the method of Lagrange inversion. Generating functions derived in these three ways are amenable to our asymptotic analyses, and we state some further general results that apply to these cases.\",\"bibtex\":\"@article{pemantle2008twenty,\\n  title={Twenty combinatorial examples of asymptotics derived from multivariate generating functions},\\n  author={Pemantle, Robin and Wilson, Mark C.},\\n  journal={SIAM Review},\\n  volume={50},\\n  number={2},\\n  pages={199-272},\\n  year={2008},\\n  publisher={Society for Industrial and Applied Mathematics},\\n  keywords={ACSV theory},\\n  url_Paper={https://epubs.siam.org/doi/epdf/10.1137/050643866},\\n  abstract={Let $F$ be a power series in at least two variables that defines a\\nmeromorphic function in a neighbourhood of the origin; for example, $F$\\nmay be a rational multivariate generating function. We discuss recent\\nresults that allow the effective computation of asymptotic expansions\\nfor the coefficients of  $F$, uniform in certain explicitly defined\\ncones of directions.\\n\\nThe purpose of this article is to illustrate the use of these techniques\\non a variety of problems of combinatorial interest. The first part\\nreviews the Morse-theoretic underpinnings of these techniques, and then\\nsummarizes the necessary results so that only elementary analyses are\\nneeded to check hypotheses and carry out computations. The remainder\\nfocuses on combinatorial applications. Specific examples deal with\\nenumeration of words with forbidden substrings, edges and cycles in\\ngraphs, polyominoes, descents and solutions to integer equations. After\\nthe individual examples, we discuss three broad classes of examples,\\nnamely functions derived via the transfer matrix method, those derived\\nvia the kernel method, and those derived via the method of Lagrange\\ninversion. Generating functions derived in these three ways are amenable\\nto our asymptotic analyses, and we state some further general results\\nthat apply to these cases.}\\n}\\n\\n\",\"author_short\":[\"Pemantle, R.\",\"Wilson, M. C.\"],\"key\":\"pemantle2008twenty\",\"id\":\"pemantle2008twenty\",\"bibbaseid\":\"pemantle-wilson-twentycombinatorialexamplesofasymptoticsderivedfrommultivariategeneratingfunctions-2008\",\"role\":\"author\",\"urls\":{\" paper\":\"https://epubs.siam.org/doi/epdf/10.1137/050643866\"},\"keyword\":[\"ACSV theory\"],\"metadata\":{\"authorlinks\":{\"wilson, m\":\"https://markcwilson.site/Research/ACSV/\"}},\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"title\":\"Asymptotics of coefficients of multivariate generating functions: improvements for smooth points\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Raichev\"],\"firstnames\":[\"Alexander\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Wilson\"],\"firstnames\":[\"Mark\",\"C.\"],\"suffixes\":[]}],\"journal\":\"Electronic Journal of Combinatorics\",\"pages\":\"R89-R89\",\"year\":\"2008\",\"keywords\":\"ACSV theory\",\"url_paper\":\"https://www.combinatorics.org/ojs/index.php/eljc/article/view/v15i1r89/pdf\",\"abstract\":\"Let $∑_{β∈ℕ^d} F_β x^β$ be a multivariate power series, a generating function for a combinatorial class perhaps. Assume that in a neighborhood of the origin this series represents a nonentire function $F=G/H^p$ where $G$ and $H$ are holomorphic and $p$ is a positive integer. Given a direction $α∈ℕ_+^d$ for which asymptotics are controlled by a smooth point of the singular variety $H = 0$, we compute the asymptotics of $F_{nα}$ as $n\\\\to∞$. We do this via multivariate singularity analysis and give an explicit formula for the full asymptotic expansion. This improves on earlier work of R. Pemantle and the second author, and allows for much more accurate numerical approximation, as demonstrated in our examples.\",\"bibtex\":\"@article{raichev2008asymptotics,\\n  title={Asymptotics of coefficients of multivariate generating functions: improvements for smooth points},\\n  author={Raichev, Alexander and Wilson, Mark C.},\\n  journal={Electronic Journal of Combinatorics},\\n  pages={R89-R89},\\n  year={2008},\\n  keywords={ACSV theory},\\n  url_Paper={https://www.combinatorics.org/ojs/index.php/eljc/article/view/v15i1r89/pdf},\\n  abstract={Let $\\\\sum_{\\\\beta\\\\in\\\\mathbb{N}^d} F_\\\\beta x^\\\\beta$ be a multivariate\\npower series, a generating function for a combinatorial class perhaps.\\nAssume that in a neighborhood of the origin this series represents a\\nnonentire function $F=G/H^p$ where $G$ and $H$ are holomorphic and $p$\\nis a positive integer. Given a direction $\\\\alpha\\\\in\\\\mathbb{N}_+^d$ for\\nwhich asymptotics are controlled by a smooth point of the singular\\nvariety $H = 0$, we compute the asymptotics of $F_{n\\\\alpha}$ as\\n$n\\\\to\\\\infty$. We do this via multivariate singularity analysis and give\\nan explicit formula for the full asymptotic expansion. This improves on\\nearlier work of R. Pemantle and the second author, and allows for much\\nmore accurate numerical approximation, as demonstrated in our examples.}\\n}\\n\\n\",\"author_short\":[\"Raichev, A.\",\"Wilson, M. C.\"],\"key\":\"raichev2008asymptotics\",\"id\":\"raichev2008asymptotics\",\"bibbaseid\":\"raichev-wilson-asymptoticsofcoefficientsofmultivariategeneratingfunctionsimprovementsforsmoothpoints-2008\",\"role\":\"author\",\"urls\":{\" paper\":\"https://www.combinatorics.org/ojs/index.php/eljc/article/view/v15i1r89/pdf\"},\"keyword\":[\"ACSV theory\"],\"metadata\":{\"authorlinks\":{\"wilson, m\":\"https://markcwilson.site/Research/ACSV/\"}},\"html\":\"\"},{\"bibtype\":\"inproceedings\",\"type\":\"inproceedings\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Raichev\"],\"firstnames\":[\"Alexander\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Wilson\"],\"firstnames\":[\"Mark\",\"C.\"],\"suffixes\":[]}],\"booktitle\":\"2007 Conference on Analysis of Algorithms, AofA 07\",\"title\":\"A new method for computing asymptotics of diagonal coefficients of multivariate generating functions\",\"pages\":\"439-449\",\"publisher\":\"Assoc. Discrete Math. Theor. Comput. Sci., Nancy\",\"series\":\"Discrete Math. Theor. Comput. Sci. Proc., AH\",\"abstract\":\"Let $∑_{\\\\mathbf{n}∈ℕ^d} F_\\\\mathbf{n} \\\\mathbf{x}^\\\\mathbf{n}$ be a multivariate generating function that converges in a neighborhood of the origin of $ℂ^d$. We present a new, multivariate method for computing the asymptotics of the diagonal coefficients $F_{a_1n,…,a_dn}$ and show its superiority over the standard, univariate diagonal method. Several examples are given in detail.\",\"keywords\":\"ACSV theory\",\"url_link\":\"https://dmtcs.episciences.org/3531/pdf\",\"year\":\"2007\",\"bibtex\":\"@InProceedings{RaWi2007,\\n  author    = {Raichev, Alexander and Wilson, Mark C.},\\n  booktitle = {2007 {C}onference on {A}nalysis of {A}lgorithms, {A}of{A} 07},\\n  title     = {A new method for computing asymptotics of diagonal coefficients of multivariate generating functions},\\n  pages     = {439-449},\\n  publisher = {Assoc. Discrete Math. Theor. Comput. Sci., Nancy},\\n  series    = {Discrete Math. Theor. Comput. Sci. Proc., AH},\\n  abstract  = {Let $\\\\sum_{\\\\mathbf{n}\\\\in\\\\mathbb{N}^d} F_\\\\mathbf{n}\\n\\\\mathbf{x}^\\\\mathbf{n}$ be a multivariate generating function that\\nconverges in a neighborhood of the origin of $\\\\mathbb{C}^d$. We present\\na new, multivariate method for computing the asymptotics of the diagonal\\ncoefficients $F_{a_1n,\\\\ldots,a_dn}$ and show its superiority over the\\nstandard, univariate diagonal method. Several examples are given in\\ndetail.},\\n  keywords  = {ACSV theory},\\n  url_link  = {https://dmtcs.episciences.org/3531/pdf},\\n  year      = {2007},\\n}\\n\\n\",\"author_short\":[\"Raichev, A.\",\"Wilson, M. C.\"],\"key\":\"RaWi2007\",\"id\":\"RaWi2007\",\"bibbaseid\":\"raichev-wilson-anewmethodforcomputingasymptoticsofdiagonalcoefficientsofmultivariategeneratingfunctions-2007\",\"role\":\"author\",\"urls\":{\" link\":\"https://dmtcs.episciences.org/3531/pdf\"},\"keyword\":[\"ACSV theory\"],\"metadata\":{\"authorlinks\":{\"wilson, m\":\"https://markcwilson.site/Research/ACSV/\"}},\"html\":\"\"},{\"bibtype\":\"inproceedings\",\"type\":\"inproceedings\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Wilson\"],\"firstnames\":[\"Mark\",\"C.\"],\"suffixes\":[]}],\"title\":\"Asymptotics for generalized Riordan arrays\",\"booktitle\":\"2005 International Conference on Analysis of Algorithms\",\"series\":\"Discrete Math. Theor. Comput. Sci. Proc., AD\",\"pages\":\"323-333 (electronic)\",\"publisher\":\"Assoc. Discrete Math. Theor. Comput. Sci., Nancy\",\"year\":\"2005\",\"keywords\":\"ACSV theory\",\"url_paper\":\"https://dmtcs.episciences.org/3389/pdf\",\"abstract\":\"The machinery of Riordan arrays has been used recently by several authors. We show how meromorphic singularity analysis can be used to provide uniform bivariate asymptotic expansions, in the central regime, for a generalization of these arrays. We show how to do this systematically, for various descriptions of the array. Several examples from recent literature are given.\",\"bibtex\":\"@inproceedings {Wils2005,\\n    AUTHOR = {Wilson, Mark C.},\\n     TITLE = {Asymptotics for generalized {R}iordan arrays},\\n BOOKTITLE = {2005 {I}nternational {C}onference on {A}nalysis of\\n              {A}lgorithms},\\n    SERIES = {Discrete Math. Theor. Comput. Sci. Proc., AD},\\n     PAGES = {323-333 (electronic)},\\n PUBLISHER = {Assoc. Discrete Math. Theor. Comput. Sci., Nancy},\\n      YEAR = {2005},\\n       keywords={ACSV theory},\\n  url_Paper={https://dmtcs.episciences.org/3389/pdf},\\n  abstract={The machinery of Riordan arrays has been used recently by several\\nauthors. We show how meromorphic singularity analysis can be used to\\nprovide uniform bivariate asymptotic expansions, in the central regime,\\nfor a generalization of these arrays. We show how to do this\\nsystematically, for various descriptions of the array. Several examples\\nfrom recent literature are given.}\\n }\\n\\n\",\"author_short\":[\"Wilson, M. C.\"],\"key\":\"Wils2005\",\"id\":\"Wils2005\",\"bibbaseid\":\"wilson-asymptoticsforgeneralizedriordanarrays-2005\",\"role\":\"author\",\"urls\":{\" paper\":\"https://dmtcs.episciences.org/3389/pdf\"},\"keyword\":[\"ACSV theory\"],\"metadata\":{\"authorlinks\":{\"wilson, m\":\"https://markcwilson.site/Research/ACSV/\"}},\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"title\":\"Asymptotics of multivariate sequences II: multiple points of the singular variety\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Pemantle\"],\"firstnames\":[\"Robin\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Wilson\"],\"firstnames\":[\"Mark\",\"C.\"],\"suffixes\":[]}],\"journal\":\"Combinatorics, Probability and Computing\",\"volume\":\"13\",\"number\":\"4-5\",\"pages\":\"735-761\",\"year\":\"2004\",\"publisher\":\"Cambridge University Press\",\"keywords\":\"ACSV theory\",\"url_paper\":\"https://www.cambridge.org/core/journals/combinatorics-probability-and-computing/article/abs/asymptotics-of-multivariate-sequences-ii-multiple-points-of-the-singular-variety/F8B144EBED2754E43361A3BE4B0C3B01\",\"abstract\":\"Let $F(\\\\b{z})=∑_\\\\b{r} a_\\\\b{r}\\\\b{z^r}$ be a multivariate generating function which is meromorphic in some neighborhood of the origin of $ℂ^d$, and let $\\\\mathcal{V}$ be its set of singularities. Effective asymptotic expansions for the coefficients can be obtained by complex contour integration near points of $\\\\mathcal{V}$. In the first article in this series, we treated the case of smooth points of $\\\\mathcal{V}$. In this article we deal with multiple points of $\\\\mathcal{V}$. Our results show that the central limit (Ornstein-Zernike) behavior typical of the smooth case does not hold in the multiple point case. For example, when $\\\\mathcal{V}$ has a multiple point singularity at $(1 , … , 1)$, rather than $a_\\\\b{r}$ decaying as $|\\\\b{r}|^{-1/2}$ as $|\\\\b{r}| \\\\to ∞$, $a_\\\\b{r}$ is very nearly polynomial in a cone of directions.\",\"bibtex\":\"@article{pemantle2004asymptotics,\\n  title={Asymptotics of multivariate sequences II: multiple points of the singular variety},\\n  author={Pemantle, Robin and Wilson, Mark C.},\\n  journal={Combinatorics, Probability and Computing},\\n  volume={13},\\n  number={4-5},\\n  pages={735-761},\\n  year={2004},\\n  publisher={Cambridge University Press},\\n  keywords={ACSV theory},\\n  url_Paper={https://www.cambridge.org/core/journals/combinatorics-probability-and-computing/article/abs/asymptotics-of-multivariate-sequences-ii-multiple-points-of-the-singular-variety/F8B144EBED2754E43361A3BE4B0C3B01},\\n  abstract={Let $F(\\\\b{z})=\\\\sum_\\\\b{r} a_\\\\b{r}\\\\b{z^r}$ be a multivariate  generating\\nfunction which is meromorphic in some neighborhood of the origin of\\n$\\\\mathbb{C}^d$, and let $\\\\mathcal{V}$ be its set of singularities.\\nEffective asymptotic expansions for the coefficients can be obtained by\\ncomplex contour integration near points of $\\\\mathcal{V}$. In the first\\narticle in this series, we treated the case of smooth points  of\\n$\\\\mathcal{V}$. In this article we deal with multiple points of\\n$\\\\mathcal{V}$. Our results show that the central limit\\n(Ornstein-Zernike) behavior typical of the smooth case does not hold in\\nthe multiple point case. For example, when $\\\\mathcal{V}$ has a multiple\\npoint singularity at $(1 , \\\\ldots , 1)$, rather than $a_\\\\b{r}$ decaying\\nas $|\\\\b{r}|^{-1/2}$ as $|\\\\b{r}| \\\\to \\\\infty$, $a_\\\\b{r}$ is very nearly\\npolynomial in a cone of directions.}\\n}\\n\\n\",\"author_short\":[\"Pemantle, R.\",\"Wilson, M. C.\"],\"key\":\"pemantle2004asymptotics\",\"id\":\"pemantle2004asymptotics\",\"bibbaseid\":\"pemantle-wilson-asymptoticsofmultivariatesequencesiimultiplepointsofthesingularvariety-2004\",\"role\":\"author\",\"urls\":{\" paper\":\"https://www.cambridge.org/core/journals/combinatorics-probability-and-computing/article/abs/asymptotics-of-multivariate-sequences-ii-multiple-points-of-the-singular-variety/F8B144EBED2754E43361A3BE4B0C3B01\"},\"keyword\":[\"ACSV theory\"],\"metadata\":{\"authorlinks\":{\"wilson, m\":\"https://markcwilson.site/Research/ACSV/\"}},\"html\":\"\"},{\"bibtype\":\"article\",\"type\":\"article\",\"title\":\"Asymptotics of multivariate sequences: I. smooth points of the singular variety\",\"author\":[{\"propositions\":[],\"lastnames\":[\"Pemantle\"],\"firstnames\":[\"Robin\"],\"suffixes\":[]},{\"propositions\":[],\"lastnames\":[\"Wilson\"],\"firstnames\":[\"Mark\",\"C.\"],\"suffixes\":[]}],\"journal\":\"Journal of Combinatorial Theory, Series A\",\"volume\":\"97\",\"number\":\"1\",\"pages\":\"129-161\",\"year\":\"2002\",\"publisher\":\"Academic Press\",\"keywords\":\"ACSV theory\",\"url_paper\":\"https://www.sciencedirect.com/science/article/pii/S0097316501932017\",\"abstract\":\"Given a multivariate generating function $F(z_1 , … , z_d) = ∑ a_{r_1 , … , r_d} z_1^{r_1} ⋯ z_d^{r_d}$, we determine asymptotics for the coefficients. Our approach is to use Cauchy's integral formula near singular points of $F$, resulting in a tractable oscillating integral. This paper treats the case where the singular point of $F$ is a smooth point of a surface of poles. Companion papers G treat singular points of $F$ where the local geometry is more complicated, and for which other methods of analysis are not known.\",\"bibtex\":\"@article{pemantle2002asymptotics,\\n  title={Asymptotics of multivariate sequences: I. smooth points of the singular variety},\\n  author={Pemantle, Robin and Wilson, Mark C.},\\n  journal={Journal of Combinatorial Theory, Series A},\\n  volume={97},\\n  number={1},\\n  pages={129-161},\\n  year={2002},\\n  publisher={Academic Press},\\n  keywords={ACSV theory},\\n  url_Paper={https://www.sciencedirect.com/science/article/pii/S0097316501932017},\\n  abstract={Given a multivariate generating function $F(z_1 , \\\\ldots , z_d) = \\\\sum\\na_{r_1 , \\\\ldots , r_d} z_1^{r_1} \\\\cdots z_d^{r_d}$, we determine\\nasymptotics for the coefficients. Our approach is to use Cauchy's\\nintegral formula near singular points of $F$, resulting in a tractable\\noscillating integral. This paper treats the case where the singular\\npoint of $F$ is a smooth point of a surface of poles. Companion papers G\\ntreat singular points of $F$ where the local geometry is more\\ncomplicated, and for which other methods of analysis are not known.}\\n}\\n\\n\",\"author_short\":[\"Pemantle, R.\",\"Wilson, M. C.\"],\"key\":\"pemantle2002asymptotics\",\"id\":\"pemantle2002asymptotics\",\"bibbaseid\":\"pemantle-wilson-asymptoticsofmultivariatesequencesismoothpointsofthesingularvariety-2002\",\"role\":\"author\",\"urls\":{\" paper\":\"https://www.sciencedirect.com/science/article/pii/S0097316501932017\"},\"keyword\":[\"ACSV theory\"],\"metadata\":{\"authorlinks\":{\"wilson, m\":\"https://markcwilson.site/Research/ACSV/\"}},\"html\":\"\"}],\n      metadata: {\"owner\":\"wilson, m\",\"authorlinks\":{\"wilson, m\":\"https://markcwilson.site/Research/ACSV/\"}},\n      ownerID: \"fNyuMAbkm7wGFjWan\"\n    };\n  </script>\n\n  <script type=\"text/javascript\">\n  var site$;\n  if (typeof $ != 'undefined') {\n    console.log('existing jquery: storing and then restoring');\n    site$ = $;\n    $ = null;\n  }\n  </script>\n\n  <script src=\"https://ajax.googleapis.com/ajax/libs/jquery/2.2.4/jquery.min.js\">\n  </script>\n  <script type=\"text/javascript\">\n  if (site$) {\n    console.log('existing jquery: avoiding conflict and restoring');\n    bb$ = $.noConflict(true);\n    $ = site$;\n  } else {\n    bb$ = $;\n  }\n  </script>\n\n  <script src=\"//bibbase.org/js/bibbase.min.js\"\n          type=\"text/javascript\">\n  </script>\n\n  <script type=\"text/javascript\"\n          src=\"https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML\">\n  </script>\n  <script type=\"text/x-mathjax-config\">\n    MathJax.Hub.Config({tex2jax: {inlineMath: [['$','$'], ['\\\\(','\\\\)']]},\n                        skipTags: [\"body\"],\n                        processClass: \"bibbase_paper\"});\n  </script>\n\n  <style>\n  @media print {\n    .dontprint {\n        display: none !important;\n    }\n\n  .bibbase_group {\n    background-color: #E0E0E0;\n    font-style: italic;\n    font-size: larger;\n    text-shadow: 0px 0px 3px #FFFFFF;\n    border-radius: 4px 4px 4px 4px;\n    -moz-border-radius: 4px 4px 4px 4px;\n    -webkit-border-radius: 4px;\n    padding: 5px 40px 5px;\n    margin-top: 10px;\n    margin-bottom: 5px;\n    cursor: pointer;\n  }\n\n  .bibbase_paper {\n    margin-bottom: 5px;\n  }\n\n  }\n  </style>\n\n  \n  <div style=\"float: right; margin-right: 10px; margin-top: 10px; text-align: right; font-size: 12px\">\n    generated by\n    <a href=\"//bibbase.org\"\n       onclick=\"trackOutboundLink('bibbase', 'logo clicked', window.location.href, '//bibbase.org'); return false;\">\n      <img src=\"//bibbase.org/img/logo.svg\"\n           style=\"vertical-align: middle; margin-left: 3px; height: 16px;\"/\n           alt=\"bibbase.org\"\n           title=\"BibBase – The easiest way to maintain your publications page.\"\n        ></a>\n    \n  </div>\n  \n\n  <div id=\"bibbase_header\" class=\"dontprint\" style=\"\">\n\n    <ul class=\"nav nav-pills\" style=\"margin: 0px;\">\n\n      <li class=\"dropdown\" id=\"groupby_dropdown\">\n      <a class=\"dropdown-toggle\"\n         data-toggle=\"dropdown\"\n         href=\"#\">\n          Group by\n          <b class=\"caret\"></b>\n      </a>\n      <ul class=\"dropdown-menu\">\n        <li class=\"groupby year\"><a onclick=\"groupby('year')\">\n            Year</a>\n        </li>\n        <li class=\"groupby author\"><a onclick=\"groupby('author_short')\">\n            Author</a>\n        </li>\n        <li class=\"groupby type\"><a onclick=\"groupby('type')\">\n            Type</a>\n        </li>\n        <li class=\"groupby keyword\"><a onclick=\"groupby('keyword')\">\n            Keyword</a>\n        </li>\n        <li class=\"groupby downloads\"><a onclick=\"groupby('downloads')\">\n            Downloads</a>\n        </li>\n      </ul>\n      </li>\n\n\n      <li class=\"dropdown\" id=\"menu_dropdown\">\n      <a class=\"dropdown-toggle\"\n         data-toggle=\"dropdown\"\n         href=\"#\" alt=\"controls\">\n          <i class=\"fa fa-reorder\" alt=\"controls\"></i>\n          <b class=\"caret\"></b>\n      </a>\n      <ul class=\"dropdown-menu\">\n        <li>\n          <a onclick=\"toggleAll()\">\n            <i class=\"fa fa-chevron-down fa-fw\"></i> Expand/Collapse All\n          </a>\n        </li>\n        <li>\n          <a download=\"uc\" href=\"data:text/bibtex;base64,@Article{GMRW2022,
author = {Greenwood, Torin and Melczer, Stephen and Ruza, Tiadora and Wilson, Mark C.},
journal = {S\'{e}minaire Lotharingien de Combinatoire},
abstract = {We present a strategy for computing asymptotics of coefficients of
$d$-variate algebraic generating functions. Using known constructions,
we embed the coefficient array into an array represented by a rational
generating functions in $d+1$ variables, and then apply ACSV theory to
analyse the latter. This method allows us to give systematic results in
the multivariate case, seems more promising than trying to derive
analogs of the rational ACSV theory for algebraic GFs, and gives the
prospect of further improvements as embedding methods are studied in
more detail.},
title = {Asymptotics of coefficients of algebraic series via embedding into rational series (extended abstract)},
year = {2022},
pages = {12pp},
volume = {86B}, 
keywords={ACSV applications, ACSV theory},
url_paper = {https://www.emis.de/journals/SLC/wpapers/FPSAC2022/30.pdf},
url_slides = {},
}



@article{ramgoolam2020quiver,
  title={Quiver asymptotics: free chiral ring},
  author={Ramgoolam, Sanjaye and Wilson, Mark C and Zahabi, Ali},
  journal={Journal of Physics A: Mathematical and Theoretical},
  volume={53},
  number={10},
  pages={105401},
  year={2020},
  publisher={IOP Publishing},
  keywords={ACSV applications},
  url_Paper={https://iopscience.iop.org/article/10.1088/1751-8121/ab6fc6/pdf},
  abstract={The large N generating functions for the counting of chiral operators in
$\mathcal{N} = 1$, four-dimensional quiver gauge theories have
previously been obtained in terms of the weighted adjacency matrix of
the quiver diagram. We introduce the methods of multi-variate asymptotic
analysis to study this counting in the limit of large charges. We
describe a Hagedorn phase transition associated with this asymptotics,
which refines and generalizes known results on the 2-matrix harmonic
oscillator. Explicit results are obtained for two infinite classes of
quiver theories, namely the generalized clover quivers and affine
$\mathbb{C}^3 \setminus \hat{A}_n$ orbifold quivers.}
}





@Article{melczer2019higher,
  author     = {Melczer, Stephen and Wilson, Mark C.},
  title      = {Higher dimensional lattice walks: Connecting combinatorial and analytic behavior},
  number     = {4},
  pages      = {2140-2174},
  volume     = {33},
  abstract   = {We consider the enumeration of walks on the non-negative lattice
$\mathbb{N}^d$, with steps defined by a set $\mathcal{S} \subset \{ -1,
0, 1 \}^d \setminus \{\mathbf{0}\}$. Previous work in this area has
established asymptotics for the number of walks in certain families of
models by applying the techniques of analytic combinatorics in several
variables (ACSV), where one encodes the generating function of a lattice
path model as the diagonal of a multivariate rational function. Melczer
and Mishna obtained asymptotics when the set of steps $\mathcal{S}$ is
symmetric over every axis; in this setting one can always apply the
methods of ACSV to a multivariate rational function whose  set of
singularities is a smooth manifold (the simplest case). Here we go
further, providing asymptotics for models with generating functions that
must be encoded by multivariate rational functions having non-smooth
singular sets.  In the process, our analysis connects past work to
deeper structural results in the theory of analytic combinatorics in
several variables.  One application is a closed form for asymptotics of
models defined by step sets that are symmetric over all but one axis. As
a special case, we apply our results when $d=2$ to give a rigorous proof
of asymptotics conjectured by Bostan and Kauers; asymptotics for walks
returning to boundary axes and the origin are also given.},
  journal    = {SIAM Journal on Discrete Mathematics},
  keywords   = {ACSV applications},
  publisher  = {Society for Industrial and Applied Mathematics},
  url_paper  = {https://arxiv.org/abs/1810.06170},
  url_slides = {https://www.youtube.com/watch?v=o5SOHJbGO4g},
  year       = {2019},
}



@inproceedings {MeWi2016, 
AUTHOR = {Stephen Melczer and Wilson, Mark C.},
     TITLE = {Asymptotics of lattice walks via analytic combinatorics in several variables},
 BOOKTITLE = {2016 {C}onference on {F}ormal {P}ower {S}eries and {A}lgebraic {C}ombinatorics, FPSAC2016},
    SERIES = {Discrete Math. Theor. Comput. Sci. Proc., AH},
     PAGES = {863-874},
 PUBLISHER = {Assoc. Discrete Math. Theor. Comput. Sci., Nancy},
      YEAR = {2016},
  keywords={ACSV applications},
  url_Paper={https://dmtcs.episciences.org/6390/pdf},
  abstract={We consider the enumeration of walks on the two dimensional non-negative
integer lattice with short steps. Up to isomorphism there are 79 unique
two dimensional models to consider, and previous work in this area has
used the kernel method, along with a rigorous computer algebra approach,
to show that 23 of the 79 models admit D-finite generating functions. In
2009, Bostan and Kauers used Pad\'{e}-Hermite approximants to guess
differential equations which these 23 generating functions satisfy, in
the process guessing asymptotics of their coefficient sequences. In this
article we provide, for the first time, a complete rigorous verification
of these guesses. Our technique is to use the kernel method to express
19 of the 23 generating functions as diagonals of tri-variate rational
functions and apply the methods of analytic combinatorics in several
variables (the remaining 4 models have algebraic generating functions
and can thus be handled by univariate techniques). This approach also
shows the link between combinatorial properties of the models and
features of its asymptotics such as asymptotic and polynomial growth
factors. In addition, we give expressions for the number of walks
returning to the x-axis, the y-axis, and the origin, proving recently
conjectured asymptotics of Bostan, Chyzak, van Hoeij, Kauers, and Pech.}
  
}




@article{wilson2015diagonal,
  title={Diagonal asymptotics for products of combinatorial classes},
  author={Wilson, Mark C.},
  journal={Combinatorics, Probability and Computing},
  volume={24},
  number={1},
  pages={354-372},
  year={2015},
  publisher={Cambridge University Press},
  keywords={ACSV theory},
  url_Paper={https://www.cambridge.org/core/journals/combinatorics-probability-and-computing/article/abs/diagonal-asymptotics-for-products-of-combinatorial-classes/DE65A4AA078A6161905DD6CEFBEDC85E},
  abstract={We generalize and improve recent results by B\'{o}na and Knopfmacher and
by Banderier and Hitczenko concerning the joint distribution of the sum
and number of parts in tuples of restricted compositions. Specifically,
we generalize the problem to general combinatorial classes and relax the
requirement that the sizes of the compositions be equal. We extend the
main explicit results to enumeration problems whose counting sequences
are Riordan arrays.  In this framework, we give an alternative method
for computing asymptotics in the supercritical case, which avoids
explicit diagonal extraction and seems likely to be computationally more
efficient.}
}



@book{pemantle2013analytic,
  title={Analytic combinatorics in several variables},
  author={Pemantle, Robin and Wilson, Mark C.},
  year={2013},
  publisher={Cambridge University Press},
  keywords={ACSV theory},
  url_Paper={},
  abstract={}
}



@article{raichev2012new,
  title={A new approach to asymptotics of Maclaurin coefficients of algebraic functions},
  author={Raichev, Alexander and Wilson, Mark C.},
  journal={arXiv preprint arXiv:1202.3826},
  year={2012},
  keywords={ACSV theory},
  url_Paper={https://arxiv.org/pdf/1202.3826.pdf},
  abstract={We propose a general method for deriving asymptotics of the Maclaurin
series coefficients of algebraic functions that is based on a procedure of K. V. Safonov
and multivariate singularity analysis. We test the feasibility of this this approach by
experimenting on several examples.}
}



@article{raichev2010asymptotics,
  title={Asymptotics of coefficients of multivariate generating functions: improvements for multiple points},
  author={Raichev, Alexander and Wilson, Mark C.},
  journal={arXiv preprint arXiv:1009.5715},
  year={2010},
  keywords={ACSV theory},
  url_Paper={},
  abstract={Let $F(x)= \sum_{\nu\in\mathbb{N}^d} F_\nu x^\nu$ be a multivariate
power series with complex coefficients that converges in a neighborhood
of the origin. Assume $F=G/H$ for some functions $G$ and $H$ holomorphic
in a neighborhood of the origin. We derive asymptotics for the
coefficients $F_{r\alpha}$ as $r \to \infty$ with $r\alpha \in
\mathbb{N}^d$ for $\alpha$ in a permissible subset of $d$-tuples of
positive reals. More specifically, we give an algorithm for computing
arbitrary terms of the asymptotic expansion for $F_{r\alpha}$ when the
asymptotics are controlled by a transverse multiple point of the
analytic variety $H = 0$. This improves upon earlier work  by R.
Pemantle and M. C. Wilson. We have implemented our algorithm in Sage and
apply it to obtain accurate numerical results for several rational
combinatorial generating functions.}
}



@incollection {PeWi2010,
    AUTHOR = {Pemantle, Robin and Wilson, Mark C.},
     TITLE = {Asymptotic expansions of oscillatory integrals with complex
              phase},
 BOOKTITLE = {Algorithmic probability and combinatorics},
    SERIES = {Contemp. Math.},
    VOLUME = {520},
     PAGES = {221-240},
 PUBLISHER = {Amer. Math. Soc., Providence, RI},
      YEAR = {2010},
       keywords={ACSV theory},
  url_Paper={https://arxiv.org/pdf/0903.3585.pdf},
  abstract={We consider saddle point integrals in $d$ variables whose phase function
is neither real nor purely imaginary. Results analogous to those for
Laplace (real phase) and Fourier (imaginary phase) integrals hold
whenever the phase function is analytic and nondegenerate. These results
generalize what is well known for integrals of Laplace and Fourier type.
The method is via contour shifting in complex $d$-space. This work is
motivated by applications to asymptotic enumeration.}
}



@article{pemantle2008twenty,
  title={Twenty combinatorial examples of asymptotics derived from multivariate generating functions},
  author={Pemantle, Robin and Wilson, Mark C.},
  journal={SIAM Review},
  volume={50},
  number={2},
  pages={199-272},
  year={2008},
  publisher={Society for Industrial and Applied Mathematics},
  keywords={ACSV theory},
  url_Paper={https://epubs.siam.org/doi/epdf/10.1137/050643866},
  abstract={Let $F$ be a power series in at least two variables that defines a
meromorphic function in a neighbourhood of the origin; for example, $F$
may be a rational multivariate generating function. We discuss recent
results that allow the effective computation of asymptotic expansions
for the coefficients of  $F$, uniform in certain explicitly defined
cones of directions.

The purpose of this article is to illustrate the use of these techniques
on a variety of problems of combinatorial interest. The first part
reviews the Morse-theoretic underpinnings of these techniques, and then
summarizes the necessary results so that only elementary analyses are
needed to check hypotheses and carry out computations. The remainder
focuses on combinatorial applications. Specific examples deal with
enumeration of words with forbidden substrings, edges and cycles in
graphs, polyominoes, descents and solutions to integer equations. After
the individual examples, we discuss three broad classes of examples,
namely functions derived via the transfer matrix method, those derived
via the kernel method, and those derived via the method of Lagrange
inversion. Generating functions derived in these three ways are amenable
to our asymptotic analyses, and we state some further general results
that apply to these cases.}
}



@article{raichev2008asymptotics,
  title={Asymptotics of coefficients of multivariate generating functions: improvements for smooth points},
  author={Raichev, Alexander and Wilson, Mark C.},
  journal={Electronic Journal of Combinatorics},
  pages={R89-R89},
  year={2008},
  keywords={ACSV theory},
  url_Paper={https://www.combinatorics.org/ojs/index.php/eljc/article/view/v15i1r89/pdf},
  abstract={Let $\sum_{\beta\in\mathbb{N}^d} F_\beta x^\beta$ be a multivariate
power series, a generating function for a combinatorial class perhaps.
Assume that in a neighborhood of the origin this series represents a
nonentire function $F=G/H^p$ where $G$ and $H$ are holomorphic and $p$
is a positive integer. Given a direction $\alpha\in\mathbb{N}_+^d$ for
which asymptotics are controlled by a smooth point of the singular
variety $H = 0$, we compute the asymptotics of $F_{n\alpha}$ as
$n\to\infty$. We do this via multivariate singularity analysis and give
an explicit formula for the full asymptotic expansion. This improves on
earlier work of R. Pemantle and the second author, and allows for much
more accurate numerical approximation, as demonstrated in our examples.}
}



@InProceedings{RaWi2007,
  author    = {Raichev, Alexander and Wilson, Mark C.},
  booktitle = {2007 {C}onference on {A}nalysis of {A}lgorithms, {A}of{A} 07},
  title     = {A new method for computing asymptotics of diagonal coefficients of multivariate generating functions},
  pages     = {439-449},
  publisher = {Assoc. Discrete Math. Theor. Comput. Sci., Nancy},
  series    = {Discrete Math. Theor. Comput. Sci. Proc., AH},
  abstract  = {Let $\sum_{\mathbf{n}\in\mathbb{N}^d} F_\mathbf{n}
\mathbf{x}^\mathbf{n}$ be a multivariate generating function that
converges in a neighborhood of the origin of $\mathbb{C}^d$. We present
a new, multivariate method for computing the asymptotics of the diagonal
coefficients $F_{a_1n,\ldots,a_dn}$ and show its superiority over the
standard, univariate diagonal method. Several examples are given in
detail.},
  keywords  = {ACSV theory},
  url_link  = {https://dmtcs.episciences.org/3531/pdf},
  year      = {2007},
}



@inproceedings {Wils2005,
    AUTHOR = {Wilson, Mark C.},
     TITLE = {Asymptotics for generalized {R}iordan arrays},
 BOOKTITLE = {2005 {I}nternational {C}onference on {A}nalysis of
              {A}lgorithms},
    SERIES = {Discrete Math. Theor. Comput. Sci. Proc., AD},
     PAGES = {323-333 (electronic)},
 PUBLISHER = {Assoc. Discrete Math. Theor. Comput. Sci., Nancy},
      YEAR = {2005},
       keywords={ACSV theory},
  url_Paper={https://dmtcs.episciences.org/3389/pdf},
  abstract={The machinery of Riordan arrays has been used recently by several
authors. We show how meromorphic singularity analysis can be used to
provide uniform bivariate asymptotic expansions, in the central regime,
for a generalization of these arrays. We show how to do this
systematically, for various descriptions of the array. Several examples
from recent literature are given.}
 }



@article{pemantle2004asymptotics,
  title={Asymptotics of multivariate sequences II: multiple points of the singular variety},
  author={Pemantle, Robin and Wilson, Mark C.},
  journal={Combinatorics, Probability and Computing},
  volume={13},
  number={4-5},
  pages={735-761},
  year={2004},
  publisher={Cambridge University Press},
  keywords={ACSV theory},
  url_Paper={https://www.cambridge.org/core/journals/combinatorics-probability-and-computing/article/abs/asymptotics-of-multivariate-sequences-ii-multiple-points-of-the-singular-variety/F8B144EBED2754E43361A3BE4B0C3B01},
  abstract={Let $F(\b{z})=\sum_\b{r} a_\b{r}\b{z^r}$ be a multivariate  generating
function which is meromorphic in some neighborhood of the origin of
$\mathbb{C}^d$, and let $\mathcal{V}$ be its set of singularities.
Effective asymptotic expansions for the coefficients can be obtained by
complex contour integration near points of $\mathcal{V}$. In the first
article in this series, we treated the case of smooth points  of
$\mathcal{V}$. In this article we deal with multiple points of
$\mathcal{V}$. Our results show that the central limit
(Ornstein-Zernike) behavior typical of the smooth case does not hold in
the multiple point case. For example, when $\mathcal{V}$ has a multiple
point singularity at $(1 , \ldots , 1)$, rather than $a_\b{r}$ decaying
as $|\b{r}|^{-1/2}$ as $|\b{r}| \to \infty$, $a_\b{r}$ is very nearly
polynomial in a cone of directions.}
}



@article{pemantle2002asymptotics,
  title={Asymptotics of multivariate sequences: I. smooth points of the singular variety},
  author={Pemantle, Robin and Wilson, Mark C.},
  journal={Journal of Combinatorial Theory, Series A},
  volume={97},
  number={1},
  pages={129-161},
  year={2002},
  publisher={Academic Press},
  keywords={ACSV theory},
  url_Paper={https://www.sciencedirect.com/science/article/pii/S0097316501932017},
  abstract={Given a multivariate generating function $F(z_1 , \ldots , z_d) = \sum
a_{r_1 , \ldots , r_d} z_1^{r_1} \cdots z_d^{r_d}$, we determine
asymptotics for the coefficients. Our approach is to use Cauchy's
integral formula near singular points of $F$, resulting in a tractable
oscillating integral. This paper treats the case where the singular
point of $F$ is a smooth point of a surface of poles. Companion papers G
treat singular points of $F$ where the local geometry is more
complicated, and for which other methods of analysis are not known.}
}

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C.</a>\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://www.emis.de/journals/SLC/wpapers/FPSAC2022/30.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'greenwood-melczer-ruza-wilson-asymptoticsofcoefficientsofalgebraicseriesviaembeddingintorationalseriesextendedabstract-2022', 'https://www.emis.de/journals/SLC/wpapers/FPSAC2022/30.pdf', event);\">\n    Asymptotics of coefficients of algebraic series via embedding into rational series (extended abstract).\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Séminaire Lotharingien de Combinatoire</i>, 86B: 12pp.  2022.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://www.emis.de/journals/SLC/wpapers/FPSAC2022/30.pdf\"\n     onclick=\"log_download('greenwood-melczer-ruza-wilson-asymptoticsofcoefficientsofalgebraicseriesviaembeddingintorationalseriesextendedabstract-2022', 'https://www.emis.de/journals/SLC/wpapers/FPSAC2022/30.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"Asymptotics of coefficients of algebraic series via embedding into rational series (extended abstract) [pdf]\"\n       title=\" paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> paper</span></a>\n  &nbsp;\n  \n  <a href=\"https://drive.google.com/uc?export=download&amp;id=1NEXsxwRAx2CWt43v0y0jyZdP1LdOS_xF\"\n     onclick=\"log_download('greenwood-melczer-ruza-wilson-asymptoticsofcoefficientsofalgebraicseriesviaembeddingintorationalseriesextendedabstract-2022', 'https://drive.google.com/uc?export=download&amp;id=1NEXsxwRAx2CWt43v0y0jyZdP1LdOS_xF', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Asymptotics of coefficients of algebraic series via embedding into rational series (extended abstract) [link]\"\n       title=\" slides\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> slides</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/greenwood-melczer-ruza-wilson-asymptoticsofcoefficientsofalgebraicseriesviaembeddingintorationalseriesextendedabstract-2022\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>4 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('greenwood-melczer-ruza-wilson-asymptoticsofcoefficientsofalgebraicseriesviaembeddingintorationalseriesextendedabstract-2022')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@Article{GMRW2022,\nauthor = {Greenwood, Torin and Melczer, Stephen and Ruza, Tiadora and Wilson, Mark C.},\njournal = {S\\&#x27;{e}minaire Lotharingien de Combinatoire},\nabstract = {We present a strategy for computing asymptotics of coefficients of\n$d$-variate algebraic generating functions. Using known constructions,\nwe embed the coefficient array into an array represented by a rational\ngenerating functions in $d+1$ variables, and then apply ACSV theory to\nanalyse the latter. This method allows us to give systematic results in\nthe multivariate case, seems more promising than trying to derive\nanalogs of the rational ACSV theory for algebraic GFs, and gives the\nprospect of further improvements as embedding methods are studied in\nmore detail.},\ntitle = {Asymptotics of coefficients of algebraic series via embedding into rational series (extended abstract)},\nyear = {2022},\npages = {12pp},\nvolume = {86B}, \nkeywords={ACSV applications, ACSV theory},\nurl_paper = {https://www.emis.de/journals/SLC/wpapers/FPSAC2022/30.pdf},\nurl_slides = {},\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We present a strategy for computing asymptotics of coefficients of $d$-variate algebraic generating functions. Using known constructions, we embed the coefficient array into an array represented by a rational generating functions in $d+1$ variables, and then apply ACSV theory to analyse the latter. This method allows us to give systematic results in the multivariate case, seems more promising than trying to derive analogs of the rational ACSV theory for algebraic GFs, and gives the prospect of further improvements as embedding methods are studied in more detail.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2020\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2020')\"\n      onkeypress=\"toggleGroup('group_2020')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2020</span>\n      <span class=\"bibbase_group_count\">\n        \n        (1)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"ramgoolam-wilson-zahabi-quiverasymptoticsfreechiralring-2020\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n  Ramgoolam, S.; <a class=\"bibbase author link\" href=\"https://markcwilson.site/Research/ACSV/\">Wilson, M. C</a>;  and Zahabi, A.\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://iopscience.iop.org/article/10.1088/1751-8121/ab6fc6/pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'ramgoolam-wilson-zahabi-quiverasymptoticsfreechiralring-2020', 'https://iopscience.iop.org/article/10.1088/1751-8121/ab6fc6/pdf', event);\">\n    Quiver asymptotics: free chiral ring.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Journal of Physics A: Mathematical and Theoretical</i>, 53(10): 105401.  2020.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://iopscience.iop.org/article/10.1088/1751-8121/ab6fc6/pdf\"\n     onclick=\"log_download('ramgoolam-wilson-zahabi-quiverasymptoticsfreechiralring-2020', 'https://iopscience.iop.org/article/10.1088/1751-8121/ab6fc6/pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Quiver asymptotics: free chiral ring [link]\"\n       title=\" paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> paper</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/ramgoolam-wilson-zahabi-quiverasymptoticsfreechiralring-2020\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('ramgoolam-wilson-zahabi-quiverasymptoticsfreechiralring-2020')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{ramgoolam2020quiver,\n  title={Quiver asymptotics: free chiral ring},\n  author={Ramgoolam, Sanjaye and Wilson, Mark C and Zahabi, Ali},\n  journal={Journal of Physics A: Mathematical and Theoretical},\n  volume={53},\n  number={10},\n  pages={105401},\n  year={2020},\n  publisher={IOP Publishing},\n  keywords={ACSV applications},\n  url_Paper={https://iopscience.iop.org/article/10.1088/1751-8121/ab6fc6/pdf},\n  abstract={The large N generating functions for the counting of chiral operators in\n$\\mathcal{N} = 1$, four-dimensional quiver gauge theories have\npreviously been obtained in terms of the weighted adjacency matrix of\nthe quiver diagram. We introduce the methods of multi-variate asymptotic\nanalysis to study this counting in the limit of large charges. We\ndescribe a Hagedorn phase transition associated with this asymptotics,\nwhich refines and generalizes known results on the 2-matrix harmonic\noscillator. Explicit results are obtained for two infinite classes of\nquiver theories, namely the generalized clover quivers and affine\n$\\mathbb{C}^3 \\setminus \\hat{A}_n$ orbifold quivers.}\n}\n\n\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  The large N generating functions for the counting of chiral operators in $\\mathcal{N} = 1$, four-dimensional quiver gauge theories have previously been obtained in terms of the weighted adjacency matrix of the quiver diagram. We introduce the methods of multi-variate asymptotic analysis to study this counting in the limit of large charges. We describe a Hagedorn phase transition associated with this asymptotics, which refines and generalizes known results on the 2-matrix harmonic oscillator. Explicit results are obtained for two infinite classes of quiver theories, namely the generalized clover quivers and affine $ℂ^3 ∖ Â_n$ orbifold quivers.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2019\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2019')\"\n      onkeypress=\"toggleGroup('group_2019')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2019</span>\n      <span class=\"bibbase_group_count\">\n        \n        (1)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"melczer-wilson-higherdimensionallatticewalksconnectingcombinatorialandanalyticbehavior-2019\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n  Melczer, S.;  and <a class=\"bibbase author link\" href=\"https://markcwilson.site/Research/ACSV/\">Wilson, M. C.</a>\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/abs/1810.06170\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'melczer-wilson-higherdimensionallatticewalksconnectingcombinatorialandanalyticbehavior-2019', 'https://arxiv.org/abs/1810.06170', event);\">\n    Higher dimensional lattice walks: Connecting combinatorial and analytic behavior.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>SIAM Journal on Discrete Mathematics</i>, 33(4): 2140-2174.  2019.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/abs/1810.06170\"\n     onclick=\"log_download('melczer-wilson-higherdimensionallatticewalksconnectingcombinatorialandanalyticbehavior-2019', 'https://arxiv.org/abs/1810.06170', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Higher dimensional lattice walks: Connecting combinatorial and analytic behavior [link]\"\n       title=\" paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> paper</span></a>\n  &nbsp;\n  \n  <a href=\"https://www.youtube.com/watch?v=o5SOHJbGO4g\"\n     onclick=\"log_download('melczer-wilson-higherdimensionallatticewalksconnectingcombinatorialandanalyticbehavior-2019', 'https://www.youtube.com/watch?v=o5SOHJbGO4g', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Higher dimensional lattice walks: Connecting combinatorial and analytic behavior [link]\"\n       title=\" slides\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> slides</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/melczer-wilson-higherdimensionallatticewalksconnectingcombinatorialandanalyticbehavior-2019\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('melczer-wilson-higherdimensionallatticewalksconnectingcombinatorialandanalyticbehavior-2019')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@Article{melczer2019higher,\n  author     = {Melczer, Stephen and Wilson, Mark C.},\n  title      = {Higher dimensional lattice walks: Connecting combinatorial and analytic behavior},\n  number     = {4},\n  pages      = {2140-2174},\n  volume     = {33},\n  abstract   = {We consider the enumeration of walks on the non-negative lattice\n$\\mathbb{N}^d$, with steps defined by a set $\\mathcal{S} \\subset \\{ -1,\n0, 1 \\}^d \\setminus \\{\\mathbf{0}\\}$. Previous work in this area has\nestablished asymptotics for the number of walks in certain families of\nmodels by applying the techniques of analytic combinatorics in several\nvariables (ACSV), where one encodes the generating function of a lattice\npath model as the diagonal of a multivariate rational function. Melczer\nand Mishna obtained asymptotics when the set of steps $\\mathcal{S}$ is\nsymmetric over every axis; in this setting one can always apply the\nmethods of ACSV to a multivariate rational function whose  set of\nsingularities is a smooth manifold (the simplest case). Here we go\nfurther, providing asymptotics for models with generating functions that\nmust be encoded by multivariate rational functions having non-smooth\nsingular sets.  In the process, our analysis connects past work to\ndeeper structural results in the theory of analytic combinatorics in\nseveral variables.  One application is a closed form for asymptotics of\nmodels defined by step sets that are symmetric over all but one axis. As\na special case, we apply our results when $d=2$ to give a rigorous proof\nof asymptotics conjectured by Bostan and Kauers; asymptotics for walks\nreturning to boundary axes and the origin are also given.},\n  journal    = {SIAM Journal on Discrete Mathematics},\n  keywords   = {ACSV applications},\n  publisher  = {Society for Industrial and Applied Mathematics},\n  url_paper  = {https://arxiv.org/abs/1810.06170},\n  url_slides = {https://www.youtube.com/watch?v=o5SOHJbGO4g},\n  year       = {2019},\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We consider the enumeration of walks on the non-negative lattice $ℕ^d$, with steps defined by a set $\\mathcal{S} ⊂ \\{ -1, 0, 1 \\}^d ∖ \\{\\mathbf{0}\\}$. Previous work in this area has established asymptotics for the number of walks in certain families of models by applying the techniques of analytic combinatorics in several variables (ACSV), where one encodes the generating function of a lattice path model as the diagonal of a multivariate rational function. Melczer and Mishna obtained asymptotics when the set of steps $\\mathcal{S}$ is symmetric over every axis; in this setting one can always apply the methods of ACSV to a multivariate rational function whose set of singularities is a smooth manifold (the simplest case). Here we go further, providing asymptotics for models with generating functions that must be encoded by multivariate rational functions having non-smooth singular sets. In the process, our analysis connects past work to deeper structural results in the theory of analytic combinatorics in several variables. One application is a closed form for asymptotics of models defined by step sets that are symmetric over all but one axis. As a special case, we apply our results when $d=2$ to give a rigorous proof of asymptotics conjectured by Bostan and Kauers; asymptotics for walks returning to boundary axes and the origin are also given.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2016\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2016')\"\n      onkeypress=\"toggleGroup('group_2016')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2016</span>\n      <span class=\"bibbase_group_count\">\n        \n        (1)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"melczer-wilson-asymptoticsoflatticewalksviaanalyticcombinatoricsinseveralvariables-2016\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n  Melczer, S.;  and <a class=\"bibbase author link\" href=\"https://markcwilson.site/Research/ACSV/\">Wilson, M. C.</a>\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://dmtcs.episciences.org/6390/pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'melczer-wilson-asymptoticsoflatticewalksviaanalyticcombinatoricsinseveralvariables-2016', 'https://dmtcs.episciences.org/6390/pdf', event);\">\n    Asymptotics of lattice walks via analytic combinatorics in several variables.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  In <i>2016 Conference on Formal Power Series and Algebraic Combinatorics, FPSAC2016</i>, of <i>Discrete Math. Theor. Comput. Sci. Proc., AH</i>, pages 863-874,  2016. Assoc. Discrete Math. Theor. Comput. Sci., Nancy\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://dmtcs.episciences.org/6390/pdf\"\n     onclick=\"log_download('melczer-wilson-asymptoticsoflatticewalksviaanalyticcombinatoricsinseveralvariables-2016', 'https://dmtcs.episciences.org/6390/pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Asymptotics of lattice walks via analytic combinatorics in several variables [link]\"\n       title=\" paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> paper</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/melczer-wilson-asymptoticsoflatticewalksviaanalyticcombinatoricsinseveralvariables-2016\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('melczer-wilson-asymptoticsoflatticewalksviaanalyticcombinatoricsinseveralvariables-2016')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@inproceedings {MeWi2016, \nAUTHOR = {Stephen Melczer and Wilson, Mark C.},\n     TITLE = {Asymptotics of lattice walks via analytic combinatorics in several variables},\n BOOKTITLE = {2016 {C}onference on {F}ormal {P}ower {S}eries and {A}lgebraic {C}ombinatorics, FPSAC2016},\n    SERIES = {Discrete Math. Theor. Comput. Sci. Proc., AH},\n     PAGES = {863-874},\n PUBLISHER = {Assoc. Discrete Math. Theor. Comput. Sci., Nancy},\n      YEAR = {2016},\n  keywords={ACSV applications},\n  url_Paper={https://dmtcs.episciences.org/6390/pdf},\n  abstract={We consider the enumeration of walks on the two dimensional non-negative\ninteger lattice with short steps. Up to isomorphism there are 79 unique\ntwo dimensional models to consider, and previous work in this area has\nused the kernel method, along with a rigorous computer algebra approach,\nto show that 23 of the 79 models admit D-finite generating functions. In\n2009, Bostan and Kauers used Pad\\&#x27;{e}-Hermite approximants to guess\ndifferential equations which these 23 generating functions satisfy, in\nthe process guessing asymptotics of their coefficient sequences. In this\narticle we provide, for the first time, a complete rigorous verification\nof these guesses. Our technique is to use the kernel method to express\n19 of the 23 generating functions as diagonals of tri-variate rational\nfunctions and apply the methods of analytic combinatorics in several\nvariables (the remaining 4 models have algebraic generating functions\nand can thus be handled by univariate techniques). This approach also\nshows the link between combinatorial properties of the models and\nfeatures of its asymptotics such as asymptotic and polynomial growth\nfactors. In addition, we give expressions for the number of walks\nreturning to the x-axis, the y-axis, and the origin, proving recently\nconjectured asymptotics of Bostan, Chyzak, van Hoeij, Kauers, and Pech.}\n  \n}\n\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We consider the enumeration of walks on the two dimensional non-negative integer lattice with short steps. Up to isomorphism there are 79 unique two dimensional models to consider, and previous work in this area has used the kernel method, along with a rigorous computer algebra approach, to show that 23 of the 79 models admit D-finite generating functions. In 2009, Bostan and Kauers used Padé-Hermite approximants to guess differential equations which these 23 generating functions satisfy, in the process guessing asymptotics of their coefficient sequences. In this article we provide, for the first time, a complete rigorous verification of these guesses. Our technique is to use the kernel method to express 19 of the 23 generating functions as diagonals of tri-variate rational functions and apply the methods of analytic combinatorics in several variables (the remaining 4 models have algebraic generating functions and can thus be handled by univariate techniques). This approach also shows the link between combinatorial properties of the models and features of its asymptotics such as asymptotic and polynomial growth factors. In addition, we give expressions for the number of walks returning to the x-axis, the y-axis, and the origin, proving recently conjectured asymptotics of Bostan, Chyzak, van Hoeij, Kauers, and Pech.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2015\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2015')\"\n      onkeypress=\"toggleGroup('group_2015')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2015</span>\n      <span class=\"bibbase_group_count\">\n        \n        (1)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"wilson-diagonalasymptoticsforproductsofcombinatorialclasses-2015\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://markcwilson.site/Research/ACSV/\">Wilson, M. C.</a>\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://www.cambridge.org/core/journals/combinatorics-probability-and-computing/article/abs/diagonal-asymptotics-for-products-of-combinatorial-classes/DE65A4AA078A6161905DD6CEFBEDC85E\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'wilson-diagonalasymptoticsforproductsofcombinatorialclasses-2015', 'https://www.cambridge.org/core/journals/combinatorics-probability-and-computing/article/abs/diagonal-asymptotics-for-products-of-combinatorial-classes/DE65A4AA078A6161905DD6CEFBEDC85E', event);\">\n    Diagonal asymptotics for products of combinatorial classes.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Combinatorics, Probability and Computing</i>, 24(1): 354-372.  2015.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://www.cambridge.org/core/journals/combinatorics-probability-and-computing/article/abs/diagonal-asymptotics-for-products-of-combinatorial-classes/DE65A4AA078A6161905DD6CEFBEDC85E\"\n     onclick=\"log_download('wilson-diagonalasymptoticsforproductsofcombinatorialclasses-2015', 'https://www.cambridge.org/core/journals/combinatorics-probability-and-computing/article/abs/diagonal-asymptotics-for-products-of-combinatorial-classes/DE65A4AA078A6161905DD6CEFBEDC85E', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Diagonal asymptotics for products of combinatorial classes [link]\"\n       title=\" paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> paper</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/wilson-diagonalasymptoticsforproductsofcombinatorialclasses-2015\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('wilson-diagonalasymptoticsforproductsofcombinatorialclasses-2015')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{wilson2015diagonal,\n  title={Diagonal asymptotics for products of combinatorial classes},\n  author={Wilson, Mark C.},\n  journal={Combinatorics, Probability and Computing},\n  volume={24},\n  number={1},\n  pages={354-372},\n  year={2015},\n  publisher={Cambridge University Press},\n  keywords={ACSV theory},\n  url_Paper={https://www.cambridge.org/core/journals/combinatorics-probability-and-computing/article/abs/diagonal-asymptotics-for-products-of-combinatorial-classes/DE65A4AA078A6161905DD6CEFBEDC85E},\n  abstract={We generalize and improve recent results by B\\&#x27;{o}na and Knopfmacher and\nby Banderier and Hitczenko concerning the joint distribution of the sum\nand number of parts in tuples of restricted compositions. Specifically,\nwe generalize the problem to general combinatorial classes and relax the\nrequirement that the sizes of the compositions be equal. We extend the\nmain explicit results to enumeration problems whose counting sequences\nare Riordan arrays.  In this framework, we give an alternative method\nfor computing asymptotics in the supercritical case, which avoids\nexplicit diagonal extraction and seems likely to be computationally more\nefficient.}\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We generalize and improve recent results by Bóna and Knopfmacher and by Banderier and Hitczenko concerning the joint distribution of the sum and number of parts in tuples of restricted compositions. Specifically, we generalize the problem to general combinatorial classes and relax the requirement that the sizes of the compositions be equal. We extend the main explicit results to enumeration problems whose counting sequences are Riordan arrays. In this framework, we give an alternative method for computing asymptotics in the supercritical case, which avoids explicit diagonal extraction and seems likely to be computationally more efficient.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2013\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2013')\"\n      onkeypress=\"toggleGroup('group_2013')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2013</span>\n      <span class=\"bibbase_group_count\">\n        \n        (1)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"pemantle-wilson-analyticcombinatoricsinseveralvariables-2013\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n  Pemantle, R.;  and <a class=\"bibbase author link\" href=\"https://markcwilson.site/Research/ACSV/\">Wilson, M. C.</a>\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://drive.google.com/uc?export=download&amp;id=1NEXsxwRAx2CWt43v0y0jyZdP1LdOS_xF\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'pemantle-wilson-analyticcombinatoricsinseveralvariables-2013', 'https://drive.google.com/uc?export=download&amp;id=1NEXsxwRAx2CWt43v0y0jyZdP1LdOS_xF', event);\">\n    Analytic combinatorics in several variables.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  Cambridge University Press,  2013.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://drive.google.com/uc?export=download&amp;id=1NEXsxwRAx2CWt43v0y0jyZdP1LdOS_xF\"\n     onclick=\"log_download('pemantle-wilson-analyticcombinatoricsinseveralvariables-2013', 'https://drive.google.com/uc?export=download&amp;id=1NEXsxwRAx2CWt43v0y0jyZdP1LdOS_xF', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Analytic combinatorics in several variables [link]\"\n       title=\" paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> paper</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/pemantle-wilson-analyticcombinatoricsinseveralvariables-2013\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n\n  \n  &nbsp;\n  <span class=\"bibbase_stats_paper\" style=\"color: #777;\">\n    <span>2 downloads</span>\n  </span>\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('pemantle-wilson-analyticcombinatoricsinseveralvariables-2013')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@book{pemantle2013analytic,\n  title={Analytic combinatorics in several variables},\n  author={Pemantle, Robin and Wilson, Mark C.},\n  year={2013},\n  publisher={Cambridge University Press},\n  keywords={ACSV theory},\n  url_Paper={},\n  abstract={}\n}\n\n</pre>\n</div>\n\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2012\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2012')\"\n      onkeypress=\"toggleGroup('group_2012')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2012</span>\n      <span class=\"bibbase_group_count\">\n        \n        (1)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"raichev-wilson-anewapproachtoasymptoticsofmaclaurincoefficientsofalgebraicfunctions-2012\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n  Raichev, A.;  and <a class=\"bibbase author link\" href=\"https://markcwilson.site/Research/ACSV/\">Wilson, M. C.</a>\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/pdf/1202.3826.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'raichev-wilson-anewapproachtoasymptoticsofmaclaurincoefficientsofalgebraicfunctions-2012', 'https://arxiv.org/pdf/1202.3826.pdf', event);\">\n    A new approach to asymptotics of Maclaurin coefficients of algebraic functions.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>arXiv preprint arXiv:1202.3826</i>.  2012.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/pdf/1202.3826.pdf\"\n     onclick=\"log_download('raichev-wilson-anewapproachtoasymptoticsofmaclaurincoefficientsofalgebraicfunctions-2012', 'https://arxiv.org/pdf/1202.3826.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"A new approach to asymptotics of Maclaurin coefficients of algebraic functions [pdf]\"\n       title=\" paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> paper</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/raichev-wilson-anewapproachtoasymptoticsofmaclaurincoefficientsofalgebraicfunctions-2012\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('raichev-wilson-anewapproachtoasymptoticsofmaclaurincoefficientsofalgebraicfunctions-2012')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{raichev2012new,\n  title={A new approach to asymptotics of Maclaurin coefficients of algebraic functions},\n  author={Raichev, Alexander and Wilson, Mark C.},\n  journal={arXiv preprint arXiv:1202.3826},\n  year={2012},\n  keywords={ACSV theory},\n  url_Paper={https://arxiv.org/pdf/1202.3826.pdf},\n  abstract={We propose a general method for deriving asymptotics of the Maclaurin\nseries coefficients of algebraic functions that is based on a procedure of K. V. Safonov\nand multivariate singularity analysis. We test the feasibility of this this approach by\nexperimenting on several examples.}\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We propose a general method for deriving asymptotics of the Maclaurin series coefficients of algebraic functions that is based on a procedure of K. V. Safonov and multivariate singularity analysis. We test the feasibility of this this approach by experimenting on several examples.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2010\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2010')\"\n      onkeypress=\"toggleGroup('group_2010')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2010</span>\n      <span class=\"bibbase_group_count\">\n        \n        (2)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"raichev-wilson-asymptoticsofcoefficientsofmultivariategeneratingfunctionsimprovementsformultiplepoints-2010\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n  Raichev, A.;  and <a class=\"bibbase author link\" href=\"https://markcwilson.site/Research/ACSV/\">Wilson, M. C.</a>\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://drive.google.com/uc?export=download&amp;id=1NEXsxwRAx2CWt43v0y0jyZdP1LdOS_xF\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'raichev-wilson-asymptoticsofcoefficientsofmultivariategeneratingfunctionsimprovementsformultiplepoints-2010', 'https://drive.google.com/uc?export=download&amp;id=1NEXsxwRAx2CWt43v0y0jyZdP1LdOS_xF', event);\">\n    Asymptotics of coefficients of multivariate generating functions: improvements for multiple points.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>arXiv preprint arXiv:1009.5715</i>.  2010.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://drive.google.com/uc?export=download&amp;id=1NEXsxwRAx2CWt43v0y0jyZdP1LdOS_xF\"\n     onclick=\"log_download('raichev-wilson-asymptoticsofcoefficientsofmultivariategeneratingfunctionsimprovementsformultiplepoints-2010', 'https://drive.google.com/uc?export=download&amp;id=1NEXsxwRAx2CWt43v0y0jyZdP1LdOS_xF', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Asymptotics of coefficients of multivariate generating functions: improvements for multiple points [link]\"\n       title=\" paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> paper</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/raichev-wilson-asymptoticsofcoefficientsofmultivariategeneratingfunctionsimprovementsformultiplepoints-2010\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('raichev-wilson-asymptoticsofcoefficientsofmultivariategeneratingfunctionsimprovementsformultiplepoints-2010')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{raichev2010asymptotics,\n  title={Asymptotics of coefficients of multivariate generating functions: improvements for multiple points},\n  author={Raichev, Alexander and Wilson, Mark C.},\n  journal={arXiv preprint arXiv:1009.5715},\n  year={2010},\n  keywords={ACSV theory},\n  url_Paper={},\n  abstract={Let $F(x)= \\sum_{\\nu\\in\\mathbb{N}^d} F_\\nu x^\\nu$ be a multivariate\npower series with complex coefficients that converges in a neighborhood\nof the origin. Assume $F=G/H$ for some functions $G$ and $H$ holomorphic\nin a neighborhood of the origin. We derive asymptotics for the\ncoefficients $F_{r\\alpha}$ as $r \\to \\infty$ with $r\\alpha \\in\n\\mathbb{N}^d$ for $\\alpha$ in a permissible subset of $d$-tuples of\npositive reals. More specifically, we give an algorithm for computing\narbitrary terms of the asymptotic expansion for $F_{r\\alpha}$ when the\nasymptotics are controlled by a transverse multiple point of the\nanalytic variety $H = 0$. This improves upon earlier work  by R.\nPemantle and M. C. Wilson. We have implemented our algorithm in Sage and\napply it to obtain accurate numerical results for several rational\ncombinatorial generating functions.}\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  Let $F(x)= ∑_{ν∈ℕ^d} F_ν x^ν$ be a multivariate power series with complex coefficients that converges in a neighborhood of the origin. Assume $F=G/H$ for some functions $G$ and $H$ holomorphic in a neighborhood of the origin. We derive asymptotics for the coefficients $F_{rα}$ as $r \\to ∞$ with $rα ∈ ℕ^d$ for $α$ in a permissible subset of $d$-tuples of positive reals. More specifically, we give an algorithm for computing arbitrary terms of the asymptotic expansion for $F_{rα}$ when the asymptotics are controlled by a transverse multiple point of the analytic variety $H = 0$. This improves upon earlier work by R. Pemantle and M. C. Wilson. We have implemented our algorithm in Sage and apply it to obtain accurate numerical results for several rational combinatorial generating functions.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"pemantle-wilson-asymptoticexpansionsofoscillatoryintegralswithcomplexphase-2010\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n  Pemantle, R.;  and <a class=\"bibbase author link\" href=\"https://markcwilson.site/Research/ACSV/\">Wilson, M. C.</a>\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://arxiv.org/pdf/0903.3585.pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'pemantle-wilson-asymptoticexpansionsofoscillatoryintegralswithcomplexphase-2010', 'https://arxiv.org/pdf/0903.3585.pdf', event);\">\n    Asymptotic expansions of oscillatory integrals with complex phase.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  In <i>Algorithmic probability and combinatorics</i>, volume 520, of Contemp. Math., pages 221-240. Amer. Math. Soc., Providence, RI,  2010.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://arxiv.org/pdf/0903.3585.pdf\"\n     onclick=\"log_download('pemantle-wilson-asymptoticexpansionsofoscillatoryintegralswithcomplexphase-2010', 'https://arxiv.org/pdf/0903.3585.pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/pdf.svg\"\n\t     alt=\"Asymptotic expansions of oscillatory integrals with complex phase [pdf]\"\n       title=\" paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> paper</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/pemantle-wilson-asymptoticexpansionsofoscillatoryintegralswithcomplexphase-2010\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('pemantle-wilson-asymptoticexpansionsofoscillatoryintegralswithcomplexphase-2010')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@incollection {PeWi2010,\n    AUTHOR = {Pemantle, Robin and Wilson, Mark C.},\n     TITLE = {Asymptotic expansions of oscillatory integrals with complex\n              phase},\n BOOKTITLE = {Algorithmic probability and combinatorics},\n    SERIES = {Contemp. Math.},\n    VOLUME = {520},\n     PAGES = {221-240},\n PUBLISHER = {Amer. Math. Soc., Providence, RI},\n      YEAR = {2010},\n       keywords={ACSV theory},\n  url_Paper={https://arxiv.org/pdf/0903.3585.pdf},\n  abstract={We consider saddle point integrals in $d$ variables whose phase function\nis neither real nor purely imaginary. Results analogous to those for\nLaplace (real phase) and Fourier (imaginary phase) integrals hold\nwhenever the phase function is analytic and nondegenerate. These results\ngeneralize what is well known for integrals of Laplace and Fourier type.\nThe method is via contour shifting in complex $d$-space. This work is\nmotivated by applications to asymptotic enumeration.}\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  We consider saddle point integrals in $d$ variables whose phase function is neither real nor purely imaginary. Results analogous to those for Laplace (real phase) and Fourier (imaginary phase) integrals hold whenever the phase function is analytic and nondegenerate. These results generalize what is well known for integrals of Laplace and Fourier type. The method is via contour shifting in complex $d$-space. This work is motivated by applications to asymptotic enumeration.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2008\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2008')\"\n      onkeypress=\"toggleGroup('group_2008')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2008</span>\n      <span class=\"bibbase_group_count\">\n        \n        (2)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"pemantle-wilson-twentycombinatorialexamplesofasymptoticsderivedfrommultivariategeneratingfunctions-2008\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n  Pemantle, R.;  and <a class=\"bibbase author link\" href=\"https://markcwilson.site/Research/ACSV/\">Wilson, M. C.</a>\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://epubs.siam.org/doi/epdf/10.1137/050643866\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'pemantle-wilson-twentycombinatorialexamplesofasymptoticsderivedfrommultivariategeneratingfunctions-2008', 'https://epubs.siam.org/doi/epdf/10.1137/050643866', event);\">\n    Twenty combinatorial examples of asymptotics derived from multivariate generating functions.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>SIAM Review</i>, 50(2): 199-272.  2008.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://epubs.siam.org/doi/epdf/10.1137/050643866\"\n     onclick=\"log_download('pemantle-wilson-twentycombinatorialexamplesofasymptoticsderivedfrommultivariategeneratingfunctions-2008', 'https://epubs.siam.org/doi/epdf/10.1137/050643866', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Twenty combinatorial examples of asymptotics derived from multivariate generating functions [link]\"\n       title=\" paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> paper</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/pemantle-wilson-twentycombinatorialexamplesofasymptoticsderivedfrommultivariategeneratingfunctions-2008\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('pemantle-wilson-twentycombinatorialexamplesofasymptoticsderivedfrommultivariategeneratingfunctions-2008')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{pemantle2008twenty,\n  title={Twenty combinatorial examples of asymptotics derived from multivariate generating functions},\n  author={Pemantle, Robin and Wilson, Mark C.},\n  journal={SIAM Review},\n  volume={50},\n  number={2},\n  pages={199-272},\n  year={2008},\n  publisher={Society for Industrial and Applied Mathematics},\n  keywords={ACSV theory},\n  url_Paper={https://epubs.siam.org/doi/epdf/10.1137/050643866},\n  abstract={Let $F$ be a power series in at least two variables that defines a\nmeromorphic function in a neighbourhood of the origin; for example, $F$\nmay be a rational multivariate generating function. We discuss recent\nresults that allow the effective computation of asymptotic expansions\nfor the coefficients of  $F$, uniform in certain explicitly defined\ncones of directions.\n\nThe purpose of this article is to illustrate the use of these techniques\non a variety of problems of combinatorial interest. The first part\nreviews the Morse-theoretic underpinnings of these techniques, and then\nsummarizes the necessary results so that only elementary analyses are\nneeded to check hypotheses and carry out computations. The remainder\nfocuses on combinatorial applications. Specific examples deal with\nenumeration of words with forbidden substrings, edges and cycles in\ngraphs, polyominoes, descents and solutions to integer equations. After\nthe individual examples, we discuss three broad classes of examples,\nnamely functions derived via the transfer matrix method, those derived\nvia the kernel method, and those derived via the method of Lagrange\ninversion. Generating functions derived in these three ways are amenable\nto our asymptotic analyses, and we state some further general results\nthat apply to these cases.}\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  Let $F$ be a power series in at least two variables that defines a meromorphic function in a neighbourhood of the origin; for example, $F$ may be a rational multivariate generating function. We discuss recent results that allow the effective computation of asymptotic expansions for the coefficients of $F$, uniform in certain explicitly defined cones of directions. The purpose of this article is to illustrate the use of these techniques on a variety of problems of combinatorial interest. The first part reviews the Morse-theoretic underpinnings of these techniques, and then summarizes the necessary results so that only elementary analyses are needed to check hypotheses and carry out computations. The remainder focuses on combinatorial applications. Specific examples deal with enumeration of words with forbidden substrings, edges and cycles in graphs, polyominoes, descents and solutions to integer equations. After the individual examples, we discuss three broad classes of examples, namely functions derived via the transfer matrix method, those derived via the kernel method, and those derived via the method of Lagrange inversion. Generating functions derived in these three ways are amenable to our asymptotic analyses, and we state some further general results that apply to these cases.\n</div>\n\n\n</div>\n\n\n  <div class=\"bibbase_paper\" id=\"raichev-wilson-asymptoticsofcoefficientsofmultivariategeneratingfunctionsimprovementsforsmoothpoints-2008\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n  Raichev, A.;  and <a class=\"bibbase author link\" href=\"https://markcwilson.site/Research/ACSV/\">Wilson, M. C.</a>\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://www.combinatorics.org/ojs/index.php/eljc/article/view/v15i1r89/pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'raichev-wilson-asymptoticsofcoefficientsofmultivariategeneratingfunctionsimprovementsforsmoothpoints-2008', 'https://www.combinatorics.org/ojs/index.php/eljc/article/view/v15i1r89/pdf', event);\">\n    Asymptotics of coefficients of multivariate generating functions: improvements for smooth points.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Electronic Journal of Combinatorics</i>,R89-R89.  2008.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://www.combinatorics.org/ojs/index.php/eljc/article/view/v15i1r89/pdf\"\n     onclick=\"log_download('raichev-wilson-asymptoticsofcoefficientsofmultivariategeneratingfunctionsimprovementsforsmoothpoints-2008', 'https://www.combinatorics.org/ojs/index.php/eljc/article/view/v15i1r89/pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Asymptotics of coefficients of multivariate generating functions: improvements for smooth points [link]\"\n       title=\" paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> paper</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/raichev-wilson-asymptoticsofcoefficientsofmultivariategeneratingfunctionsimprovementsforsmoothpoints-2008\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('raichev-wilson-asymptoticsofcoefficientsofmultivariategeneratingfunctionsimprovementsforsmoothpoints-2008')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{raichev2008asymptotics,\n  title={Asymptotics of coefficients of multivariate generating functions: improvements for smooth points},\n  author={Raichev, Alexander and Wilson, Mark C.},\n  journal={Electronic Journal of Combinatorics},\n  pages={R89-R89},\n  year={2008},\n  keywords={ACSV theory},\n  url_Paper={https://www.combinatorics.org/ojs/index.php/eljc/article/view/v15i1r89/pdf},\n  abstract={Let $\\sum_{\\beta\\in\\mathbb{N}^d} F_\\beta x^\\beta$ be a multivariate\npower series, a generating function for a combinatorial class perhaps.\nAssume that in a neighborhood of the origin this series represents a\nnonentire function $F=G/H^p$ where $G$ and $H$ are holomorphic and $p$\nis a positive integer. Given a direction $\\alpha\\in\\mathbb{N}_+^d$ for\nwhich asymptotics are controlled by a smooth point of the singular\nvariety $H = 0$, we compute the asymptotics of $F_{n\\alpha}$ as\n$n\\to\\infty$. We do this via multivariate singularity analysis and give\nan explicit formula for the full asymptotic expansion. This improves on\nearlier work of R. Pemantle and the second author, and allows for much\nmore accurate numerical approximation, as demonstrated in our examples.}\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  Let $∑_{β∈ℕ^d} F_β x^β$ be a multivariate power series, a generating function for a combinatorial class perhaps. Assume that in a neighborhood of the origin this series represents a nonentire function $F=G/H^p$ where $G$ and $H$ are holomorphic and $p$ is a positive integer. Given a direction $α∈ℕ_+^d$ for which asymptotics are controlled by a smooth point of the singular variety $H = 0$, we compute the asymptotics of $F_{nα}$ as $n\\to∞$. We do this via multivariate singularity analysis and give an explicit formula for the full asymptotic expansion. This improves on earlier work of R. Pemantle and the second author, and allows for much more accurate numerical approximation, as demonstrated in our examples.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2007\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2007')\"\n      onkeypress=\"toggleGroup('group_2007')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2007</span>\n      <span class=\"bibbase_group_count\">\n        \n        (1)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"raichev-wilson-anewmethodforcomputingasymptoticsofdiagonalcoefficientsofmultivariategeneratingfunctions-2007\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n  Raichev, A.;  and <a class=\"bibbase author link\" href=\"https://markcwilson.site/Research/ACSV/\">Wilson, M. C.</a>\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://dmtcs.episciences.org/3531/pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'raichev-wilson-anewmethodforcomputingasymptoticsofdiagonalcoefficientsofmultivariategeneratingfunctions-2007', 'https://dmtcs.episciences.org/3531/pdf', event);\">\n    A new method for computing asymptotics of diagonal coefficients of multivariate generating functions.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  In <i>2007 Conference on Analysis of Algorithms, AofA 07</i>, of <i>Discrete Math. Theor. Comput. Sci. Proc., AH</i>, pages 439-449,  2007. Assoc. Discrete Math. Theor. Comput. Sci., Nancy\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://dmtcs.episciences.org/3531/pdf\"\n     onclick=\"log_download('raichev-wilson-anewmethodforcomputingasymptoticsofdiagonalcoefficientsofmultivariategeneratingfunctions-2007', 'https://dmtcs.episciences.org/3531/pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"A new method for computing asymptotics of diagonal coefficients of multivariate generating functions [link]\"\n       title=\" link\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> link</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/raichev-wilson-anewmethodforcomputingasymptoticsofdiagonalcoefficientsofmultivariategeneratingfunctions-2007\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('raichev-wilson-anewmethodforcomputingasymptoticsofdiagonalcoefficientsofmultivariategeneratingfunctions-2007')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@InProceedings{RaWi2007,\n  author    = {Raichev, Alexander and Wilson, Mark C.},\n  booktitle = {2007 {C}onference on {A}nalysis of {A}lgorithms, {A}of{A} 07},\n  title     = {A new method for computing asymptotics of diagonal coefficients of multivariate generating functions},\n  pages     = {439-449},\n  publisher = {Assoc. Discrete Math. Theor. Comput. Sci., Nancy},\n  series    = {Discrete Math. Theor. Comput. Sci. Proc., AH},\n  abstract  = {Let $\\sum_{\\mathbf{n}\\in\\mathbb{N}^d} F_\\mathbf{n}\n\\mathbf{x}^\\mathbf{n}$ be a multivariate generating function that\nconverges in a neighborhood of the origin of $\\mathbb{C}^d$. We present\na new, multivariate method for computing the asymptotics of the diagonal\ncoefficients $F_{a_1n,\\ldots,a_dn}$ and show its superiority over the\nstandard, univariate diagonal method. Several examples are given in\ndetail.},\n  keywords  = {ACSV theory},\n  url_link  = {https://dmtcs.episciences.org/3531/pdf},\n  year      = {2007},\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  Let $∑_{\\mathbf{n}∈ℕ^d} F_\\mathbf{n} \\mathbf{x}^\\mathbf{n}$ be a multivariate generating function that converges in a neighborhood of the origin of $ℂ^d$. We present a new, multivariate method for computing the asymptotics of the diagonal coefficients $F_{a_1n,…,a_dn}$ and show its superiority over the standard, univariate diagonal method. Several examples are given in detail.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2005\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2005')\"\n      onkeypress=\"toggleGroup('group_2005')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2005</span>\n      <span class=\"bibbase_group_count\">\n        \n        (1)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"wilson-asymptoticsforgeneralizedriordanarrays-2005\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n  <a class=\"bibbase author link\" href=\"https://markcwilson.site/Research/ACSV/\">Wilson, M. C.</a>\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://dmtcs.episciences.org/3389/pdf\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'wilson-asymptoticsforgeneralizedriordanarrays-2005', 'https://dmtcs.episciences.org/3389/pdf', event);\">\n    Asymptotics for generalized Riordan arrays.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  In <i>2005 International Conference on Analysis of Algorithms</i>, of <i>Discrete Math. Theor. Comput. Sci. Proc., AD</i>, pages 323-333 (electronic),  2005. Assoc. Discrete Math. Theor. Comput. Sci., Nancy\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://dmtcs.episciences.org/3389/pdf\"\n     onclick=\"log_download('wilson-asymptoticsforgeneralizedriordanarrays-2005', 'https://dmtcs.episciences.org/3389/pdf', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Asymptotics for generalized Riordan arrays [link]\"\n       title=\" paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> paper</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/wilson-asymptoticsforgeneralizedriordanarrays-2005\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('wilson-asymptoticsforgeneralizedriordanarrays-2005')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@inproceedings {Wils2005,\n    AUTHOR = {Wilson, Mark C.},\n     TITLE = {Asymptotics for generalized {R}iordan arrays},\n BOOKTITLE = {2005 {I}nternational {C}onference on {A}nalysis of\n              {A}lgorithms},\n    SERIES = {Discrete Math. Theor. Comput. Sci. Proc., AD},\n     PAGES = {323-333 (electronic)},\n PUBLISHER = {Assoc. Discrete Math. Theor. Comput. Sci., Nancy},\n      YEAR = {2005},\n       keywords={ACSV theory},\n  url_Paper={https://dmtcs.episciences.org/3389/pdf},\n  abstract={The machinery of Riordan arrays has been used recently by several\nauthors. We show how meromorphic singularity analysis can be used to\nprovide uniform bivariate asymptotic expansions, in the central regime,\nfor a generalization of these arrays. We show how to do this\nsystematically, for various descriptions of the array. Several examples\nfrom recent literature are given.}\n }\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  The machinery of Riordan arrays has been used recently by several authors. We show how meromorphic singularity analysis can be used to provide uniform bivariate asymptotic expansions, in the central regime, for a generalization of these arrays. We show how to do this systematically, for various descriptions of the array. Several examples from recent literature are given.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2004\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2004')\"\n      onkeypress=\"toggleGroup('group_2004')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2004</span>\n      <span class=\"bibbase_group_count\">\n        \n        (1)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"pemantle-wilson-asymptoticsofmultivariatesequencesiimultiplepointsofthesingularvariety-2004\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n  Pemantle, R.;  and <a class=\"bibbase author link\" href=\"https://markcwilson.site/Research/ACSV/\">Wilson, M. C.</a>\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://www.cambridge.org/core/journals/combinatorics-probability-and-computing/article/abs/asymptotics-of-multivariate-sequences-ii-multiple-points-of-the-singular-variety/F8B144EBED2754E43361A3BE4B0C3B01\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'pemantle-wilson-asymptoticsofmultivariatesequencesiimultiplepointsofthesingularvariety-2004', 'https://www.cambridge.org/core/journals/combinatorics-probability-and-computing/article/abs/asymptotics-of-multivariate-sequences-ii-multiple-points-of-the-singular-variety/F8B144EBED2754E43361A3BE4B0C3B01', event);\">\n    Asymptotics of multivariate sequences II: multiple points of the singular variety.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Combinatorics, Probability and Computing</i>, 13(4-5): 735-761.  2004.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://www.cambridge.org/core/journals/combinatorics-probability-and-computing/article/abs/asymptotics-of-multivariate-sequences-ii-multiple-points-of-the-singular-variety/F8B144EBED2754E43361A3BE4B0C3B01\"\n     onclick=\"log_download('pemantle-wilson-asymptoticsofmultivariatesequencesiimultiplepointsofthesingularvariety-2004', 'https://www.cambridge.org/core/journals/combinatorics-probability-and-computing/article/abs/asymptotics-of-multivariate-sequences-ii-multiple-points-of-the-singular-variety/F8B144EBED2754E43361A3BE4B0C3B01', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Asymptotics of multivariate sequences II: multiple points of the singular variety [link]\"\n       title=\" paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> paper</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/pemantle-wilson-asymptoticsofmultivariatesequencesiimultiplepointsofthesingularvariety-2004\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('pemantle-wilson-asymptoticsofmultivariatesequencesiimultiplepointsofthesingularvariety-2004')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{pemantle2004asymptotics,\n  title={Asymptotics of multivariate sequences II: multiple points of the singular variety},\n  author={Pemantle, Robin and Wilson, Mark C.},\n  journal={Combinatorics, Probability and Computing},\n  volume={13},\n  number={4-5},\n  pages={735-761},\n  year={2004},\n  publisher={Cambridge University Press},\n  keywords={ACSV theory},\n  url_Paper={https://www.cambridge.org/core/journals/combinatorics-probability-and-computing/article/abs/asymptotics-of-multivariate-sequences-ii-multiple-points-of-the-singular-variety/F8B144EBED2754E43361A3BE4B0C3B01},\n  abstract={Let $F(\\b{z})=\\sum_\\b{r} a_\\b{r}\\b{z^r}$ be a multivariate  generating\nfunction which is meromorphic in some neighborhood of the origin of\n$\\mathbb{C}^d$, and let $\\mathcal{V}$ be its set of singularities.\nEffective asymptotic expansions for the coefficients can be obtained by\ncomplex contour integration near points of $\\mathcal{V}$. In the first\narticle in this series, we treated the case of smooth points  of\n$\\mathcal{V}$. In this article we deal with multiple points of\n$\\mathcal{V}$. Our results show that the central limit\n(Ornstein-Zernike) behavior typical of the smooth case does not hold in\nthe multiple point case. For example, when $\\mathcal{V}$ has a multiple\npoint singularity at $(1 , \\ldots , 1)$, rather than $a_\\b{r}$ decaying\nas $|\\b{r}|^{-1/2}$ as $|\\b{r}| \\to \\infty$, $a_\\b{r}$ is very nearly\npolynomial in a cone of directions.}\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  Let $F(\\b{z})=∑_\\b{r} a_\\b{r}\\b{z^r}$ be a multivariate generating function which is meromorphic in some neighborhood of the origin of $ℂ^d$, and let $\\mathcal{V}$ be its set of singularities. Effective asymptotic expansions for the coefficients can be obtained by complex contour integration near points of $\\mathcal{V}$. In the first article in this series, we treated the case of smooth points of $\\mathcal{V}$. In this article we deal with multiple points of $\\mathcal{V}$. Our results show that the central limit (Ornstein-Zernike) behavior typical of the smooth case does not hold in the multiple point case. For example, when $\\mathcal{V}$ has a multiple point singularity at $(1 , … , 1)$, rather than $a_\\b{r}$ decaying as $|\\b{r}|^{-1/2}$ as $|\\b{r}| \\to ∞$, $a_\\b{r}$ is very nearly polynomial in a cone of directions.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n  <div class=\"bibbase_group_whole\" id=\"group_2002\">\n    <div class=\"bibbase_group\"\n      onclick=\"toggleGroup('group_2002')\"\n      onkeypress=\"toggleGroup('group_2002')\"\n      tabindex=\"0\">\n      <i class=\"fa fa-angle-down bibbase_group_head_icon\"></i>&nbsp;\n      <span>2002</span>\n      <span class=\"bibbase_group_count\">\n        \n        (1)\n        \n      </span>\n    </div>\n    <div class=\"bibbase_group_body\">\n      \n  \n  <div class=\"bibbase_paper\" id=\"pemantle-wilson-asymptoticsofmultivariatesequencesismoothpointsofthesingularvariety-2002\">\n  <span class=\"bibbase_paper_titleauthoryear\">\n\n  \n  <span class=\"bibbase_paper_author\">\n  Pemantle, R.;  and <a class=\"bibbase author link\" href=\"https://markcwilson.site/Research/ACSV/\">Wilson, M. C.</a>\n</span>\n\n  <span class=\"bibbase_paper_title\">\n  \n  \n  \n  <a href=\"https://www.sciencedirect.com/science/article/pii/S0097316501932017\"\n     onclick=\"return trackOutboundLink('publications', 'click', 'pemantle-wilson-asymptoticsofmultivariatesequencesismoothpointsofthesingularvariety-2002', 'https://www.sciencedirect.com/science/article/pii/S0097316501932017', event);\">\n    Asymptotics of multivariate sequences: I. smooth points of the singular variety.\n  </a>\n  \n  \n  \n</span>\n\n  \n\n</span>\n\n  <i>Journal of Combinatorial Theory, Series A</i>, 97(1): 129-161.  2002.\n  <span class=\"note\"></span>\n\n<span class=\"bibbase_note\"></span>\n\n<br class=\"bibbase_paper_content\"/>\n\n<span class=\"bibbase_paper_content dontprint\">\n\n  \n  <a href=\"https://www.sciencedirect.com/science/article/pii/S0097316501932017\"\n     onclick=\"log_download('pemantle-wilson-asymptoticsofmultivariatesequencesismoothpointsofthesingularvariety-2002', 'https://www.sciencedirect.com/science/article/pii/S0097316501932017', event.ctrlKey, event)\">\n    <img src=\"//bibbase.org/img/filetypes/link.svg\"\n\t     alt=\"Asymptotics of multivariate sequences: I. smooth points of the singular variety [link]\"\n       title=\" paper\"\n\t     class=\"bibbase_icon\"\n         ><span class=\"bibbase_icon_text\"> paper</span></a>\n  &nbsp;\n  \n\n  \n\n  \n  <a href=\"//bibbase.org/network/publication/pemantle-wilson-asymptoticsofmultivariatesequencesismoothpointsofthesingularvariety-2002\"\n     class=\"bibbase bibbase_page link\">link</a>\n  &nbsp;\n  \n\n  <a href=\"#\"\n     onclick=\"showBib(event); return false;\"\n     class=\"bibbase bibtex link\">bibtex\n    <i class=\"fa fa-caret-down\"></i></a>\n\n  \n  &nbsp;\n  <a class=\"bibbase_abstract_link bibbase link\"\n     href=\"#\"\n     onclick=\"showAbstract(event); return false;\">\n    abstract <i class=\"fa fa-caret-down\"></i></a>\n  \n\n  \n\n  <!-- <a class=\"bibbase read link hidden\" -->\n  <!--    href=\"javascript:toggleRead('pemantle-wilson-asymptoticsofmultivariatesequencesismoothpointsofthesingularvariety-2002')\" -->\n  <!--    title=\"Marking this paper as read adds it to your library on BibBase.\" -->\n  <!--    > -->\n  <!--   <i class=\"fa fa-check\"></i>&nbsp; -->\n  <!--   <span>mark as read</span> -->\n  <!-- </a> -->\n\n  &nbsp;\n  \n  \n  \n  \n\n</span>\n\n<div class=\"well well-small bibbase\" data-type=\"bibtex\"\n     style=\"display:none\">\n  <pre style=\"white-space: pre-wrap;\">@article{pemantle2002asymptotics,\n  title={Asymptotics of multivariate sequences: I. smooth points of the singular variety},\n  author={Pemantle, Robin and Wilson, Mark C.},\n  journal={Journal of Combinatorial Theory, Series A},\n  volume={97},\n  number={1},\n  pages={129-161},\n  year={2002},\n  publisher={Academic Press},\n  keywords={ACSV theory},\n  url_Paper={https://www.sciencedirect.com/science/article/pii/S0097316501932017},\n  abstract={Given a multivariate generating function $F(z_1 , \\ldots , z_d) = \\sum\na_{r_1 , \\ldots , r_d} z_1^{r_1} \\cdots z_d^{r_d}$, we determine\nasymptotics for the coefficients. Our approach is to use Cauchy&#x27;s\nintegral formula near singular points of $F$, resulting in a tractable\noscillating integral. This paper treats the case where the singular\npoint of $F$ is a smooth point of a surface of poles. Companion papers G\ntreat singular points of $F$ where the local geometry is more\ncomplicated, and for which other methods of analysis are not known.}\n}\n\n</pre>\n</div>\n\n\n<div class=\"well well-small bibbase\" data-type=\"abstract\"\n     style=\"display:none\">\n  Given a multivariate generating function $F(z_1 , … , z_d) = ∑ a_{r_1 , … , r_d} z_1^{r_1} ⋯ z_d^{r_d}$, we determine asymptotics for the coefficients. Our approach is to use Cauchy's integral formula near singular points of $F$, resulting in a tractable oscillating integral. This paper treats the case where the singular point of $F$ is a smooth point of a surface of poles. Companion papers G treat singular points of $F$ where the local geometry is more complicated, and for which other methods of analysis are not known.\n</div>\n\n\n</div>\n\n\n\n\n\n    </div>\n  </div>\n\n\n\n\n  </div>\n\n\n  <!-- Modal -->\n\n  <!-- <iframe id=\"bibbase_network_frame\" -->\n  <!--         src=\"//bibbase.org/network/control\" -->\n  <!--         style=\"display: none;\"> -->\n  <!-- </iframe> -->\n\n</div>\n"}; document.write(bibbase_data.data);