Hermitian Sums of Squares Modulo Hermitian Ideals

TL;DR

This paper explores conditions under which Hermitian polynomials can be expressed as sums of squares modulo Hermitian ideals, connecting positivity conditions with operator theory and extending classical algebraic problems.

Contribution

It introduces a novel approach based on Putinar-Scheiderer ideas to establish matrix positivity conditions for Hermitian sums of squares modulo ideals.

Findings

Positivity conditions are sufficient for certain classes of Hermitian polynomials.

Connections established between Hermitian sums of squares and operator-valued Riesz-Fejer theorem.

Links made to Hermitian versions of Hilbert's 17th problem.

Abstract

In this work we study the problem of writing a Hermitian polynomial as a Hermitian sum of squares modulo a Hermitian ideal. We investigate a novel idea of Putinar-Scheiderer to obtain necessary matrix positivity conditions for Hermitian polynomials to be Hermitian sums of squares modulo Hermitian ideals. We show that the conditions are sufficient for a class of examples making a connection to the operator-valued Riesz-Fejer theorem and block Toeplitz forms. The work fits into the larger themes of Hermitian versions of Hilbert's 17-th problem and characterizations of positivity.