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Computing Certificates of Strictly Positive Polynomials in Archimedean Quadratic Modules

Weifeng Shang, Jose Abel Castellanos Joo, Chenqi Mou, Deepak Kapur

TL;DR

This work develops a constructive framework for certifying positivity of polynomials within Archimedean quadratic modules by marrying Averkov's positivity transfer with Lasserre's sum-of-squares approximation. It extends these ideas to monogenic modules and arbitrary subsets, provides a constructive Putinar-style proof, and integrates the methods into a practical algorithmic workflow. Extensive experiments show certificates can be obtained in cases where existing tools struggle, often without explicitly adding Archimedean generators. The results advance both the theoretical understanding and practical computation of Positivstellensatz certificates with potential impact on real algebraic geometry and optimization.

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.

Computing Certificates of Strictly Positive Polynomials in Archimedean Quadratic Modules

TL;DR

This work develops a constructive framework for certifying positivity of polynomials within Archimedean quadratic modules by marrying Averkov's positivity transfer with Lasserre's sum-of-squares approximation. It extends these ideas to monogenic modules and arbitrary subsets, provides a constructive Putinar-style proof, and integrates the methods into a practical algorithmic workflow. Extensive experiments show certificates can be obtained in cases where existing tools struggle, often without explicitly adding Archimedean generators. The results advance both the theoretical understanding and practical computation of Positivstellensatz certificates with potential impact on real algebraic geometry and optimization.

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 , 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.

Paper Structure

This paper contains 19 sections, 10 theorems, 30 equations, 1 figure, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quadratic modules
  4. Averkov's Lemma to aid computing certificates of Positivstellensatz theorems
  5. Extending Averkov's results
  6. Estimation of parameters epsilon, gamma, and N
  7. Generalization of Averkov's Lemma
  8. Extending Lasserre's method to a monogenic Archimedean qua-dratic module
  9. A constructive proof of Putinar's criterion
  10. Computing certificates using the extended Averkov construction and Lasserre's method
  11. Constructing a bounded polynomial with a global lower bound for applying the extended Averkov's method
  12. Algorithm for solving the certificate problem
  13. Experiments
  14. Univariate certificate problems
  15. Certificates of strictly positive polynomials
  16. ...and 4 more sections

Key Result

Theorem 2.2

þ For a quadratic module $\mathop{\mathrm{QM}}\nolimits(G)$ of $\mathbb{R}[X]$ with $G = \{g_1,\ldots,g_s\}$, $\mathop{\mathrm{QM}}\nolimits(G)$ is Archimedean if and only if there exists some $g \in \mathop{\mathrm{QM}}\nolimits(G)$ such that $\mathop{\mathrm{S}}\nolimits(g)$ is compact.

Figures (1)

  • Figure 1: Performances of implementation of Algorithm \ref{['alg:Lasserre']} for lifted Motkzin and Robinson polynomials by $\frac{1}{d}$.

Theorems & Definitions (24)

  • Definition 2.1
  • Theorem 2.2: Putinar's criterion, P93p
  • Theorem 2.3: Putinar Positivstellensatz, P93p
  • Lemma 2.4: A13C
  • proof
  • Proposition 3.1
  • proof
  • Example 3.2
  • Example 3.3
  • Theorem 3.4: G98g
  • ...and 14 more