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

TL;DR

This paper introduces a randomized algorithm for almost-uniform sampling of points on algebraic varieties over finite fields, with provably small statistical distance and polynomial runtime under certain conditions.

Contribution

It presents a novel randomized method for sampling points on algebraic varieties that achieves near-uniform distribution with polynomial complexity.

Findings

Algorithm produces points with small statistical distance from uniform

Runtime is polynomial in polynomial description and degrees

Applicable over large finite fields with constant number of polynomials

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.