Certificates of Impossibility of Hilbert-Artin Representations of a Given Degree for Definite Polynomials and Functions

TL;DR

This paper introduces a method using semidefinite programming to certify the impossibility of representing certain positive polynomials as sums-of-squares with denominators below a specific degree, providing new certificates of infeasibility.

Contribution

It develops a novel approach for certifying the impossibility of low-degree SOS representations with non-uniform denominators, extending prior impossibility results.

Findings

Certificates of impossibility for degree 2 and 4 denominators in symmetric sextics.

Demonstrated algorithm for certifying infeasibility in specific polynomial cases.

Extended the scope of impossibility certificates to non-uniform denominators.

Abstract

We deploy numerical semidefinite programming and conversion to exact rational inequalities to certify that for a positive semidefinite input polynomial or rational function, any representation as a fraction of sums-of-squares of polynomials with real coefficients must contain polynomials in the denominator of degree no less than a given input lower bound. By Artin's solution to Hilbert's 17th problems, such representations always exist for some denominator degree. Our certificates of infeasibility are based on the generalization of Farkas's Lemma to semidefinite programming. The literature has many famous examples of impossibility of SOS representability including Motzkin's, Robinson's, Choi's and Lam's polynomials, and Reznick's lower degree bounds on uniform denominators, e.g., powers of the sum-of-squares of each variable. Our work on exact certificates for positive semidefiniteness allows for non-uniform denominators, which can have lower degree and are often easier to convert to exact identities. Here we demonstrate our algorithm by computing certificates of impossibilities for an arbitrary sum-of-squares denominator of degree 2 and 4 for some symmetric sextics in 4 and 5 variables, respectively. We can also certify impossibility of base polynomials in the denominator of restricted term structure, for instance as in Landau's reduction by one less variable.