Overdetermined systems of sparse polynomial equations

TL;DR

This paper presents a quasi-linear time algorithm for determining zeros of sparse univariate polynomial systems and extends to multivariate systems under a conjecture, improving computational efficiency in algebraic geometry.

Contribution

It introduces a quasi-linear time algorithm for univariate polynomial systems and offers a conditional extension to multivariate systems assuming the Zilber conjecture.

Findings

Efficient zero set computation for univariate sparse polynomials

Conditional method for multivariate systems under Zilber conjecture

Algorithm complexity is quasi-linear in the logarithm of the degree

Abstract

We show that, for a system of univariate polynomials given in sparse encoding, we can compute a single polynomial defining the same zero set, in time quasi-linear in the logarithm of the degree. In particular, it is possible to determine whether such a system of polynomials does have a zero in time quasi-linear in the logarithm of the degree. The underlying algorithm relies on a result of Bombieri and Zannier on multiplicatively dependent points in subvarieties of an algebraic torus. We also present the following conditional partial extension to the higher dimensional setting. Assume that the effective Zilber conjecture holds. Then, for a system of multivariate polynomials given in sparse encoding, we can compute a finite collection of complete intersections outside hypersurfaces that defines the same zero set, in time quasi-linear in the logarithm of the degree.