2020/08/24 by Julian Vill, Vill, Julian
Mathematics · #Algebraic Geometry (math.AG) #Commutative Algebra and Its Applications #FOS: Mathematics #Graph theory and applications #Optimization and Control (math.OC) #Tensor decomposition and applications #math.AG #math.OC
paper · pdf · doi:10.48550/arxiv.2008.10315
19 pages
arxiv created 2020/08/24 · openalex publication_date 2020/08/24 · arxiv updated 2020/08/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let f∈Σn,2d be a sum of squares. The Gram spectrahedron of f is a compact, convex set that parametrizes all sum of squares representations of f. Let F\subseteqGram(f) be a face of its Gram spectrahedron. We are interested in upper bounds for the dimension of F. We show that this upper bound can be determined combinatorially. As it turns out, if the degree is large enough, a face realizing this bound, is a face of a Gram spectrahedron such that the form f is singular. Thus we are also interested in finding better bounds whenever the form f is smooth.