On the bad points of positive semidefinite polynomials

TL;DR

This paper investigates the nature of bad points in positive semidefinite polynomials, showing certain polynomials lack bad points while providing examples with specific bad points, thus advancing understanding of sum of squares representations.

Contribution

It proves quartic polynomials in three variables have no bad points and constructs examples with bad points, addressing open questions in the field.

Findings

Quartic polynomials in three variables never have bad points.

Examples of positive semidefinite polynomials with a bad point at the origin.

Existence of positive semidefinite polynomials with complex bad points not on the real axis.

Abstract

A bad point of a positive semidefinite real polynomial f is a point at which a pole appears in all expressions of f as a sum of squares of rational functions. We show that quartic polynomials in three variables never have bad points. We give examples of positive semidefinite polynomials with a bad point at the origin, that are nevertheless sums of squares of formal power series, answering a question of Brumfiel. We also give an example of a positive semidefinite polynomial in three variables with a complex bad point that is not real, answering a question of Scheiderer.