Separating inequalities for nonnegative polynomials that are not sums of\n squares

2012/01/19 by Sadik Iliman, Iliman, Sadik, Timo de Wolff +1
Computer Science · Mathematics · #11E20 #11E25 #14P99 #52A20 #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #FOS: Mathematics #Mathematical functions and polynomials #Polynomial and algebraic computation

paper · pdf · doi:10.48550/arxiv.1201.4061

openalex publication_date 2012/01/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Ternary sextics and quaternary quartics are the smallest cases where there\nexist nonnegative polynomials that are not sums of squares (SOS). A complete\nclassification of the difference between these cones was given by G. Blekherman\nvia analyzing the extreme rays of the corresponding dual cones. However, an\nexact computational approach in order to build separating extreme rays for\nnonnegative polynomials that are not sums of squares is a widely open problem.\nWe provide a method substantially simplifying this computation for certain\nclasses of polynomials on the boundary of the PSD cones. In particular, our\nmethod yields separating extreme rays for every nonnegative ternary sextic with\nat least seven zeros. As an application to further instances, we compute a\nrational certificate proving that the Motzkin polynomial is not SOS.\n

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