Noncommutative sums of squares

Algorithmic aspects of sums of Hermitian squares of noncommutative polynomials

This paper presents an algorithm and its implementation in the software package NCSOStools for finding sums of hermitian squares and commutators decompositions for polynomials in noncommuting variables. The algorithm is based on noncommutative analogs of the classical Gram matrix method and the Newton polytope method, which allows us to use semidefinite programming. Throughout the paper several...

Rational Sums of Hermitian Squares of Free Noncommutative Polynomials

In this paper we consider polynomials in noncommuting variables that admit sum of hermitian squares and commutators decompositions. We recall algorithms for finding decompositions of this type that are based on semidefinite programming. The main part of the article investigates how to find such decomposition with rational coefficients if the original polynomial has rational coefficients. We sho...

Semidefinite Programming and Sums of Hermitian Squares of Noncommutative Polynomials

An algorithm for finding sums of hermitian squares decompositions for polynomials in noncommuting variables is presented. The algorithm is based on the “Newton chip method”, a noncommutative analog of the classical Newton polytope method, and semidefinite programming.

Sums of distinct squares

Sums-of-Squares Formulas

The following is the extended version of my notes from my ATC talk given on June 4, 2014 at UCLA. I begin with a basic introduction to sums-of-squares formulas, and move on to giving motivation for studying these formulas and discussing some results about them over the reals. More recent techniques have made it possible to obtain similar results over arbitrary fields, and some of these are disc...