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

2025-10-22

AI Summary

This paper investigates the computational complexity of several fundamental decision problems in optimization and computational algebra, focusing on their reducibility to the Hilbert’s Nullstellensatz problem over the reals (HN_ℝ). Using polynomial-time many-one reductions, tools from algebraic geometry, and the first-order theory of real closed fields, we establish: (1) the affine polynomial projection problem is HN_𝔽-complete over any field 𝔽; (2) the sparse shift problem is HN_ℝ-hard over general fields; (3) deciding real polynomial stability, convexity, and hyperbolicity is complete for the universal theory of ℝ. These results provide the first precise complexity characterizations for these long-standing open problems—pinpointing their completeness either in the HN_ℝ class or in the universal fragment of the theory of real closed fields—and yield the first systematic complexity-theoretic framework for algebraic optimization problems.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments

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.

Problem

• Proving computational equivalence to Hilbert's Nullstellensatz for optimization problems
• Establishing complexity of affine polynomial transformations and sparse shifts
• Determining completeness of real polynomial properties in existential theory

Innovation

• Proving complexity completeness for polynomial projection problems
• Extending hardness results to fields for sparse shift problems
• Establishing exact complexity for real polynomial property verification