Hermitian analogues of Hilbert's 17-th problem

TL;DR

This paper explores Hermitian analogues of Hilbert's 17th problem, providing surveys, explicit examples, new algebraic results, and applications to CR geometry, advancing understanding of non-negative Hermitian polynomials.

Contribution

It introduces a new algebraic theorem characterizing non-negative Hermitian polynomials as quotients of squared norms and discusses novel examples and applications.

Findings

Non-negative Hermitian symmetric polynomial divides a squared norm iff it is a quotient of squared norms.

Provides explicit examples and proofs related to Hermitian analogues of Hilbert's 17th problem.

Discusses applications to CR geometry and new examples of Putinar-Scheiderer.

Abstract

We pose and discuss several Hermitian analogues of Hilbert's $17$-th problem. We survey what is known, offer many explicit examples and some proofs, and give applications to CR geometry. We prove one new algebraic theorem: a non-negative Hermitian symmetric polynomial divides a nonzero squared norm if and only if it is a quotient of squared norms. We also discuss a new example of Putinar-Scheiderer.