Almost-Uniform Sampling of Points on High-Dimensional Algebraic Varieties

Almost-Uniform Sampling of Points on High-Dimensional Algebraic Varieties. Cheraghchi, M. & Shokrollahi, A. In Proceedings of the 26th International Symposium on Theoretical Aspects of Computer Science (STACS), volume 3, of Leibniz International Proceedings in Informatics (LIPIcs), pages 277–288, 2009.

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We consider the problem of uniform sampling of points on an algebraic variety. Specifically, we develop a randomized algorithm that, given a small set of multivariate polynomials over a sufficiently large finite field, produces a common zero of the polynomials almost uniformly at random. The statistical distance between the output distribution of the algorithm and the uniform distribution on the set of common zeros is polynomially small in the field size, and the running time of the algorithm is polynomial in the description of the polynomials and their degrees provided that the number of the polynomials is a constant.

@INPROCEEDINGS{ref:conf:CS09,
  author =	 {Mahdi Cheraghchi and Amin Shokrollahi},
  title =	 {Almost-Uniform Sampling of Points on
                  High-Dimensional Algebraic Varieties},
  booktitle =	 {Proceedings of the 26th {International Symposium on
                  Theoretical Aspects of Computer Science (STACS)}},
  pages =	 {277--288},
  year =	 2009,
  series =	 {Leibniz International Proceedings in Informatics
                  (LIPIcs)},
  volume =	 3,
  url_Link =	 {http://drops.dagstuhl.de/opus/volltexte/2009/1817},
  doi =		 {10.4230/LIPIcs.STACS.2009.1817},
  keywords =	 {Uniform sampling, Algebraic varieties, Randomized
                  algorithms, Computational complexity},
  url_Paper =	 {https://arxiv.org/abs/0902.1254},
  abstract =	 {We consider the problem of uniform sampling of
                  points on an algebraic variety.  Specifically, we
                  develop a randomized algorithm that, given a small
                  set of multivariate polynomials over a sufficiently
                  large finite field, produces a common zero of the
                  polynomials almost uniformly at random. The
                  statistical distance between the output distribution
                  of the algorithm and the uniform distribution on the
                  set of common zeros is polynomially small in the
                  field size, and the running time of the algorithm is
                  polynomial in the description of the polynomials and
                  their degrees provided that the number of the
                  polynomials is a constant.  }
}

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