The Multivariate Resultant is NP-hard in any Characteristic

TL;DR

This paper proves that determining whether the multivariate resultant of a system of polynomial equations is zero is NP-hard in any characteristic, highlighting computational complexity challenges in algebraic geometry.

Contribution

It establishes NP-hardness of resultant zero testing in any characteristic and discusses complexity bounds under different field characteristics.

Findings

NP-hardness in any characteristic for resultant zero testing

Resultant testing is in AM class under GRH in characteristic zero

Upper bound remains PSPACE in positive characteristic

Abstract

The multivariate resultant is a fundamental tool of computational algebraic geometry. It can in particular be used to decide whether a system of n homogeneous equations in n variables is satisfiable (the resultant is a polynomial in the system's coefficients which vanishes if and only if the system is satisfiable). In this paper we present several NP-hardness results for testing whether a multivariate resultant vanishes, or equivalently for deciding whether a square system of homogeneous equations is satisfiable. Our main result is that testing the resultant for zero is NP-hard under deterministic reductions in any characteristic, for systems of low-degree polynomials with coefficients in the ground field (rather than in an extension). We also observe that in characteristic zero, this problem is in the Arthur-Merlin class AM if the generalized Riemann hypothesis holds true. In positive characteristic, the best upper bound remains PSPACE.