Definite determinantal representations via orthostochastic matrices. (May 2021)

• Record Type: Journal Article
• Authors: Dey, Papri
• Is Part Of: Journal of symbolic computation. Volume 104(2021)
• Page Start: 15
• Page End: 37
• Publication Date: 2021-05
• Subjects: 15A75 -- 15B10 -- 15B51 -- 90C22; Semidefinite Programming -- LMI Representable sets -- Determinantal Polynomials -- RZ Polynomials -- Orthostochastic matrices -- Exterior algebra; 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.2020.04.005
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: Definite Determinantal polynomials play a crucial role in semidefinite programming problems. Helton and Vinnikov proved that real zero (RZ) bivariate polynomials are definite determinantals. Indeed, in general, it is a difficult problem to decide whether a given polynomial is definite determinantal, and if it is, it is of paramount interest to determine a definite determinantal representation of that polynomial. We provide a necessary and sufficient condition for the existence of definite determinantal representation of a bivariate polynomial by identifying its coefficients as scalar products of two vectors where the scalar products are defined by orthostochastic matrices. This alternative condition enables us to develop a method to compute a monic symmetric/Hermitian determinantal representations for a bivariate polynomial of degree d . In addition, we propose a computational relaxation to the determinantal problem which turns into a problem of expressing the vector of coefficients of the given polynomial as convex combinations of some specified points.