Low-Rank Sum-of-Squares Representations on Varieties of Minimal Degree

2016/06/14 by Blekherman, Grigoriy, Plaumann, Daniel, Sinn, Rainer +1
#Algebraic Geometry (math.AG) #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.1606.04387

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.

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