Low-Rank Sum-of-Squares Representations on Varieties of Minimal Degree. (18th June 2017)

• Record Type: Journal Article
• Authors: Blekherman, Grigoriy; Plaumann, Daniel; Sinn, Rainer; Vinzant, Cynthia
• Is Part Of: International mathematics research notices. Volume 2019:Number 1(2019)
• Page Start: 33
• Page End: 54
• Publication Date: 2017-06-18
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnx113
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract: A celebrated result by Hilbert says that every real nonnegative ternary quartic is a sum of three squares. We show more generally that every nonnegative quadratic form on a real projective variety $X$ of minimal degree is a sum of ${\dim(X)+1}$ squares of linear forms. This strengthens one direction of a recent result due to Blekherman, Smith, and Velasco. Our upper bound is the best possible, and it implies the existence of low-rank factorizations of positive semidefinite bivariate matrix polynomials and representations of biforms as sums of few squares. We determine the number of equivalence classes of sum-of-squares representations of general quadratic forms on surfaces of minimal degree, generalizing the count for ternary quartics by Powers, Reznick, Scheiderer, and Sottile.