(Positive) Quadratic Determinantal Representations of Quartic Curves and the Robinson Polynomial

TL;DR

This paper proves that certain smooth real nonnegative quartic curves can be represented as determinants of positive semidefinite quadratic matrix pencils, but this does not extend to the Robinson polynomial.

Contribution

It establishes a new class of quadratic determinantal representations for smooth nonnegative quartic curves and identifies a key exception with the Robinson polynomial.

Findings

Every smooth real nonnegative ternary quartic admits a positive semidefinite quadratic determinantal representation.

The Robinson polynomial does not admit such a representation, answering a specific open question.

The results connect algebraic geometry with matrix theory and polynomial optimization.

Abstract

We prove that every real nonnegative ternary quartic whose complex zero set is smooth can be represented as the determinant of a symmetric matrix with quadratic entries which is everywhere positive semidefinite. We show that the corresponding statement fails for the Robinson polynomial, answering a question by Buckley and \v{S}ivic.