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Hilbert's Nullstellensatz is in the Counting Hierarchy

Robert Andrews, Abhibhav Garg, Éric Schost

TL;DR

The paper tackles the problem of deciding Hilbert's Nullstellensatz for systems of multivariate polynomials and counting their finite solutions, proving these tasks lie in the counting hierarchy CH. The authors construct a uniform, constant-depth arithmetic-circuit framework for computing the multivariate resultant, leveraging the Poisson formula and a homotopy-based implicit function approach to express solutions in a way amenable to low-depth circuits. They then transfer these circuit constructions to the boolean setting via threshold circuits, showing that evaluation can be performed in CH, and hence that HN and related radical-membership/counting problems over ${f Q}$, number fields, and finite fields admit CH (and FP^CH) algorithms. The approach yields several corollaries, including improved complexity bounds over previously known PSPACE/FPSPACE results, uniform results across fields, and applications such as radical ideal membership, dimension computation, and tensor rank estimation. Overall, the work provides a rigorous, uniform method to place a broad class of algebraic tasks within the counting hierarchy, with potential practical impact on algebraic geometry and computational complexity theory.

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.

Hilbert's Nullstellensatz is in the Counting Hierarchy

TL;DR

The paper tackles the problem of deciding Hilbert's Nullstellensatz for systems of multivariate polynomials and counting their finite solutions, proving these tasks lie in the counting hierarchy CH. The authors construct a uniform, constant-depth arithmetic-circuit framework for computing the multivariate resultant, leveraging the Poisson formula and a homotopy-based implicit function approach to express solutions in a way amenable to low-depth circuits. They then transfer these circuit constructions to the boolean setting via threshold circuits, showing that evaluation can be performed in CH, and hence that HN and related radical-membership/counting problems over , number fields, and finite fields admit CH (and FP^CH) algorithms. The approach yields several corollaries, including improved complexity bounds over previously known PSPACE/FPSPACE results, uniform results across fields, and applications such as radical ideal membership, dimension computation, and tensor rank estimation. Overall, the work provides a rigorous, uniform method to place a broad class of algebraic tasks within the counting hierarchy, with potential practical impact on algebraic geometry and computational complexity theory.

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.
Paper Structure (14 sections, 4 theorems, 10 equations)

This paper contains 14 sections, 4 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Our results
  4. Proof overview
  5. The resultant
  6. The counting hierarchy
  7. Computing the resultant in constant depth
  8. From homogeneous to affine systems
  9. Organization
  10. Preliminaries
  11. Notation and conventions
  12. Uniformity of boolean and arithmetic circuit families
  13. Integer, rational, and finite field arithmetic
  14. Uniformity of basic operations on arithmetic circuits

Key Result

theorem 1.1

Let ${\mathbb F}$ be one of the fields ${\mathbb Q}$, ${\mathbb Q}(y_1,\ldots,y_k)$, a number field ${\mathbb K}$, ${\mathbb K}(y_1, \ldots, y_k)$, the finite field ${\mathbb F}_q$, or ${\mathbb F}_q(y_1,\ldots,y_k)$. Then Hilbert's Nullstellensatz over ${\mathbb F}$ can be decided in $\CH$.

Theorems & Definitions (18)

  • theorem 1.1: see \ref{['theorem: nullstellensatz in ch']}
  • theorem 1.2: see \ref{['lemma: application radical membership']}
  • theorem 1.3: see \ref{['lemma: application tensor rank']}
  • definition 2.1
  • definition 2.2: vollmer1999
  • definition 2.3: Direct connection language
  • definition 2.4
  • remark 2.5
  • remark 2.6
  • remark 2.7
  • ...and 8 more