A parametric version of the Hilbert Nullstellensatz

Abstract

Hilbert's Nullstellensatz is a fundamental result in algebraic geometry that gives a necessary and sufficient condition for a finite collection of multivariate polynomials to have a common zero in an algebraically closed field. Associated with this result, there is the computational problem HN of determining whether a system of polynomials with coefficients in the field of rational numbers has a common zero over the field of algebraic numbers. In an influential paper, Koiran showed that HN can be determined in the polynomial hierarchy assuming the Generalised Riemann Hypothesis (GRH). More precisely, he showed that HN lies in the complexity class AM under GRH. In a later work he generalised this result by showing that the problem DIM, which asks to determine the dimension of the set of solutions of a given polynomial system, also lies in AM subject to GRH. In this paper we study the solvability of polynomial equations over arbitrary algebraically closed fields of characteristic zero. Up to isomorphism, every such field is the algebraic closure of a field of rational functions. We thus formulate a parametric version of HN, called HNP, in which the input is a system of polynomials with coefficients in a function field $\mathbb{Q}(\mathbf{x})$ and the task is to determine whether the polynomials have a common zero in the algebraic closure $\overline{\mathbb{Q}(\mathbf{x})}$. We observe that Koiran's proof that DIM lies in AM can be interpreted as a randomised polynomial-time reduction of DIM to HNP, followed by an argument that HNP lies in AM. Our main contribution is a self-contained proof that HNP lies in AM that follows the same basic idea as Koiran's argument -- namely random instantiation of the parameters -- but whose justification is purely algebraic, relying on a parametric version of Hilbert's Nullstellensatz, and avoiding recourse to semi-algebraic geometry.