Computing Certificates of Members in Archimedean Quadratic Modules in \(A[X]\) and Certifying the Emptiness in Inconsistent Monogenic Archimedean Quadratic Modules in \(A[X_1, ..., X_n]\)

Polynomials have been found to be a powerful tool over hundreds of years for modeling problems in numerous applications in science, engineering, medicine, and other domains. In the context of formal methods, polynomials arise in modeling in aerospace software and robotics, cyber-physical and hybrid systems, autonomous vehicles and controllers based on neural networks.

A quadratic module is a linear combination of polynomials in a set of generators (including the constant 1) with sum of squares polynomials as multipliers. The membership problem for a finitely generated quadratic module can be decided; however, computing a certificate exhibiting why it is nonnegative under the assumption that the generators are nonnegative, can be nontrivial.

A new symbolic algorithm is presented to compute sums of squares multipliers (certificates) to witness the membership of univariate polynomials in Archimedean quadratic modules. An algorithm is presented to compute a certificate for \(-1\) in inconsistent quadratic modules in \(A[X_1, ..., X_n]\).

Recommended citation: Castellanos Joo, Jose A.. "Computing Certificates of Members in Archimedean Quadratic Modules in A[X] and Certifying the Emptiness in Inconsistent Monogenic Archimedean Quadratic Modules in A[X_1, ..., X_n]." (2026). https://digitalrepository.unm.edu/cs_etds/144
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