Facial reduction for exact polynomial sum of squares decomposition

Abstract

We study the problem of 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.As an application, we study the problem of determining when can a rational polynomial that is a sum of squares of polynomials with real coefficients be written as sum of squares of polynomials with rational coefficients, and 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 √2) that cannot be decomposed as a rational SOS.

Two remarks on sums of squares with rational coefficients

“…We remark that Question 5.3 from [11] has recently been given a negative answer by Laplagne [10]. This question was asking whether f is Q-sos if it becomes K-sos in an odd degree extension K/Q.…”

Practical construction of positive maps which are not completely positive

“…This is because by construction, the form F δ will always have non-trivial zeros chosen in Step 1 of Algorithm 1. To tackle this, there are many facial reduction methods available to allow this rationalization for positive semi-definite G, one such reduction is presented in [17] (see also [19] for instance).…”

Two remarks on sums of squares with rational coefficients

“…For the reader's convenience we have included a brief introduction to Gram spectrahedra in Section 4, together with proofs or references for the facts that we use. We remark that Question 5.3 from [13] has recently been given a negative answer by Laplagne [11]. This question was asking whether f is Q-sos if it becomes K-sos in an odd degree extension K/Q.…”