Paper Info
Title
Facial reduction for exact polynomial sum of squares decompositions
Abstract
We develop new tools for decomposing a non-negative polynomial as an exact sum of squares (SOS) in the case where the associated semidefinite program is feasible but not strictly feasible (for example if the polynomial has real zeros). Computing symbolically roots of the original polynomial and applying facial reduction techniques, we can solve the problem algebraically or restrict to a subspace where the problem becomes strictly feasible and a numerical approximation can be rounded to an exact solution. Our motivation for studying this problem is to determine when a rational polynomial that is a sum of squares of polynomials with real coefficients can be written as sum of squares of polynomials with rational coefficients. Applying our new strategy we can answer this question for some previously unknown cases. We first prove that if f is the sum of two squares with coefficients in an algebraic extension of Q of odd degree, then it can always be decomposed as a rational SOS. For the case of more than two polynomials we provide an example of an irreducible polynomial that is the sum of three squares with coefficients in Q((3)root 2) that cannot be decomposed as a rational SOS.
Year
DOI
Venue
2020
10.1090/mcom/3476
MATHEMATICS OF COMPUTATION
Field
DocType
Volume
Applied mathematics,Polynomial,Mathematical analysis,Explained sum of squares,Mathematics,Decomposition
Journal
89
Issue
ISSN
Citations 
322
0025-5718
0
PageRank 
References 
Authors
0.34
0
1
Name
Order
Citations
PageRank
Santiago Laplagne1132.84