Problems from Optimization and Computational Algebra Equivalent to Hilbert's Nullstellensatz

TL;DR

The paper investigates the complexity of core optimization and algebraic problems by linking them to Hilbert's Nullstellensatz through downward closures like $ ext{HN}_R$. It gives a unifying set of hardness and completeness results: (i) affine polynomial projection is $ ext{HN}_{ ext{F}}$-hard for every field $ ext{F}$ (with $ ext{R}$ yielding $ ext{∃R}$-completeness), (ii) sparse shift is $ ext{HN}_{ ext{F}}$-hard for infinite fields, (iii) convexity of quartic polynomials and nonnegativity of biquadratic forms are $orall ext{R}$-complete, and (iv) hyperbolicity and real stability of quartic polynomials are $orall ext{R}$-complete. The results are established via intricate polynomial-construction reductions that broaden prior $ ext{NP}$-hardness to $ ext{∃R}$- and $orall ext{R}$-level hardness, clarifying the precise complexity of these natural problems and showing that casting them as polynomial systems is often optimal. The techniques hinge on structured reductions that amplify monomial-count effects, Bezoutian-based hyperbolicity analysis, and careful handling of field extensions to transfer hardness across representations and base fields.

Abstract

Efficient algorithms for many problems in optimization and computational algebra often arise from casting them as systems of polynomial equations. Blum, Shub, and Smale formalized this as Hilbert's Nullstellensatz Problem $HN_R$: given multivariate polynomials over a ring $R$, decide whether they have a common solution in $R$. We can also view $HN_R$ as a complexity class by taking the downward closure of the problem $HN_R$ under polynomial-time many-one reductions. In this work, we show that many important problems from optimization and algebra are complete or hard for this class. We first consider the Affine Polynomial Projection Problem: given polynomials $f,g$, does an affine projection of the variables transform $f$ into $g$? We show that this problem is at least as hard as $HN_F$ for any field $F$. Then we consider the Sparse Shift Problem: given a polynomial, can its number of monomials be reduced by an affine shift of the variables? Prior $HN_R$-hardness for this problem was known for non-field integral domains $R$, which we extend to fields. For the special case of the real field, HN captures the existential theory of the reals and its complement captures the universal theory of the reals. We prove that the problems of deciding real stability, convexity, and hyperbolicity of a given polynomial are all complete for the universal theory of the reals, thereby pinning down their exact complexity.

Theorems & Definitions (42)

• Theorem 2.1
• Remark 2.1
• Corollary 2.2
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.5
• Theorem 2.6
• Remark 2.3
• Theorem 3.1: garding