Hilbert's Nullstellensatz is in the Counting Hierarchy

TL;DR

This paper proves that deciding the existence of solutions to polynomial systems and counting solutions can be done within the counting hierarchy, improving previous complexity bounds from PSPACE and FPSPACE.

Contribution

The authors show that Hilbert's Nullstellensatz problem is in the counting hierarchy and develop constant-depth arithmetic circuits for the multivariate resultant.

Findings

Hilbert's Nullstellensatz is in the counting hierarchy.

Counting solutions to polynomial systems is polynomial-time with counting hierarchy oracle.

Constructed uniform constant-depth arithmetic circuits for the multivariate resultant.

Abstract

We show that Hilbert's Nullstellensatz, the problem of deciding if a system of multivariate polynomial equations has a solution in the algebraic closure of the underlying field, lies in the counting hierarchy. More generally, we show that the number of solutions to a system of equations can be computed in polynomial time with oracle access to the counting hierarchy. Our results hold in particular for polynomials with coefficients in either the rational numbers or a finite field. Previously, the best-known bounds on the complexities of these problems were PSPACE and FPSPACE, respectively. Our main technical contribution is the construction of a uniform family of constant-depth arithmetic circuits that compute the multivariate resultant.