@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.}
}

{"_id":"CCwokfip9tRdkx7Eq","bibbaseid":"pemantle-wilson-asymptoticsofmultivariatesequencesiimultiplepointsofthesingularvariety-2004","author_short":["Pemantle, R.","Wilson, M. C."],"bibdata":{"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":{}}},"bibtype":"article","biburl":"https://drive.google.com/uc?export=download&id=1NEXsxwRAx2CWt43v0y0jyZdP1LdOS_xF","dataSources":["yx9ivfGLzvFNsg4LH","QdFdNQZBZSvbPbp7t","o2FBsFrZMS3PxguCG","SjBh38uzaGjRinwq8","M2T285kj8vXfTkGDx","2zimNJgtWEiEM2L3J","TAyBmAZAcnenc4YET","BCqQErP8wgW48bwvt","YjweTJPHHEQ85Pems"],"keywords":["acsv theory"],"search_terms":["asymptotics","multivariate","sequences","multiple","points","singular","variety","pemantle","wilson"],"title":"Asymptotics of multivariate sequences II: multiple points of the singular variety","year":2004}