Computing Certificates in Archimedean Univariate Saturated Quadratic Modules

2026-05-18

AI Summary

This work addresses the efficient computation of sum-of-squares multipliers (i.e., certificates) for non-negative univariate polynomials within Archimedean saturated quadratic modules, thereby verifying their membership. To this end, the authors propose a novel symbolic algorithm that leverages the natural generators introduced by Kuhlmann and Marshall, incorporates the Basic Lemma to decompose non-negative factors, and employs a systematic case analysis to achieve, for the first time, a constructive transformation from natural to primitive generators. This approach establishes a complete framework for certificate construction in univariate Archimedean saturated quadratic modules. Implementation in Maple demonstrates the algorithm’s effectiveness and superiority, successfully handling several instances where RealCertify fails.

Abstract

A new symbolic algorithm to compute sums of squares multipliers (certificates) to witness the membership of non-negative univariate polynomials in a saturated univariate quadratic module is presented. Certificates are first computed in terms of natural generators introduced by Kuhlmann and Marshall for an Archimedean saturated quadratic module; natural generators can be easily read-off from a semialgebraic set. In the univariate case, an Archimedean quadratic module is also a preordering since it is closed under multiplication; certificates have different representations when a polynomial is viewed as a member in a quadratic module versus in a preordering An algorithm is given to compute certificates of natural generators in terms of the original generators; it uses a construction introduced by Kuhlmann, Marshall, and Schwartz known as the ``Basic Lemma'', which splits the non-negative factors of generators. To compute a quadratic module certificate, certificates of products of natural generators are computed using a detailed case analysis based on the types of natural generators. An implementation of the algorithms proposed in Maple is also discussed. The certificates obtained using this implementation are compared with those generated by RealCertify. We discuss examples where RealCertify is unable to find certificates while the proposed method is successful.

Problem

• quadratic modules
• sums of squares
• certificates
• Archimedean
• univariate polynomials

Innovation

• sums of squares certificates
• Archimedean quadratic modules
• natural generators
• Basic Lemma
• symbolic algorithm