Gram spectrahedra of ternary quartics. (May 2023)

• Record Type: Journal Article
• Authors: Vill, Julian
• Is Part Of: Journal of symbolic computation. Volume 116(2023)
• Page Start: 263
• Page End: 283
• Publication Date: 2023-05
• Subjects: Gram spectrahedron -- Sums of squares -- Convexity -- Faces -- Ternary quartics -- Bitangents; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2022.10.004
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: The Gram spectrahedron of a real form f ∈ R [ x _ ] 2 d parametrizes all sum of squares representations of f . It is a compact, convex, semi-algebraic set, and we study its facial structure in the case of ternary quartics, i.e. f ∈ R [ x, y, z ] 4 . We show that the Gram spectrahedron of every smooth ternary quartic has faces of dimension 2, and generically none of dimension 1, thus answering a question in Plaumann et al. (2011) about the existence of positive dimensional faces on such Gram spectrahedra. We complete the proof in Plaumann et al. (2011) showing that the so called Steiner graph of every smooth quartic is isomorphic to K 4 ∐ K 4 . Moreover, we show that the Gram spectrahedron of a generic positive semidefinite ternary quartic contains extreme points of all expected ranks.