2009/07/08 by Grigoriy Blekherman, Blekherman, Grigoriy
Computer Science · Mathematics · #14P99 #52A99 #Advanced Optimization Algorithms Research #Algebraic Geometry (math.AG) #FOS: Mathematics #Matrix Theory and Algorithms #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.0907.1339
openalex publication_date 2009/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study dimensions of the faces of the cone of nonnegative polynomials and the cone of sums of squares; we show that there are dimensional differences between corresponding faces of these cones. These dimensional gaps occur in all cases where there exist nonnegative polynomials that are not sums of squares. As either the degree or the number of variables grows the gaps become very large, asymptotically the gaps approach the full dimension of the vector space of polynomials in n variables of degree 2d. The gaps occur generically, they are not a product of selecting special faces of the cones. Using these dimensional differences we show how to derive inequalities that separate nonnegative polynomials from sums of squares; the inequalities will hold for all sums of squares, but will fail for some nonnegative polynomials.