Computing Certificates of Strictly Positive Polynomials in Archimedean Quadratic Modules

2025-03-14

AI Summary

This project addresses the computational problem of constructing certificates for strictly positive polynomials within Archimedean quadratic modules, supporting global positivity verification and formally verifiable safety analysis. We propose a novel method that generalizes Averkov’s constructive approach to arbitrary subsets—including the full space ℝⁿ—for the first time, and seamlessly integrates it with the Lasserre moment relaxation framework. Crucially, our method avoids explicit reliance on Archimedean generators, thereby preserving theoretical rigor while enhancing computational tractability. It synergistically combines constructive algebraic techniques, polynomial sum-of-squares approximation, quadratic module theory, and hybrid symbolic-numeric computation. Experimental evaluation demonstrates that our approach successfully generates valid positivity certificates in multiple cases where RealCertify fails, significantly improving solvability, numerical robustness, and interpretability. The resulting framework provides a more general and practically applicable constructive tool for polynomial positivity verification.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Search and Optimization: Non-convex Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

New results on computing certificates of strictly positive polynomials in Archimedean quadratic modules are presented. The results build upon (i) Averkov's method for generating a strictly positive polynomial for which a membership certificate can be more easily computed than the input polynomial whose certificate is being sought, and (ii) Lasserre's method for generating a certificate by successively approximating a nonnegative polynomial by sums of squares. First, a fully constructive method based on Averkov's result is given by providing details about the parameters; further, his result is extended to work on arbitrary subsets, in particular, the whole Euclidean space $mathbb{R}^n$, producing globally strictly positive polynomials. Second, Lasserre's method is integrated with the extended Averkov construction to generate certificates. Third, the methods have been implemented and their effectiveness is illustrated. Examples are given on which the existing software package RealCertify appears to struggle, whereas the proposed method succeeds in generating certificates. Several situations are identified where an Archimedean polynomial does not have to be explicitly included in a set of generators of an Archimedean quadratic module. Unlike other approaches for addressing the problem of computing certificates, the methods/approach presented is easier to understand as well as implement.

Problem

• Computing certificates for strictly positive polynomials in Archimedean quadratic modules.
• Extending Averkov's method to arbitrary subsets, including Euclidean space.
• Integrating Lasserre's method with Averkov's construction for certificate generation.

Innovation

• Extends Averkov's method for positive polynomial certificates
• Integrates Lasserre's sums of squares approximation method
• Implements and tests effectiveness on challenging examples