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

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 Šivic.

💡 Research Summary

The paper investigates determinant representations of real non‑negative ternary quartic forms with a focus on “positive quadratic” representations, i.e., representations of a homogeneous degree‑4 polynomial (F) as the determinant of a symmetric matrix (M(x_0,x_1,x_2)) whose entries are homogeneous quadrics, such that (M(a)) is positive semidefinite for every real point (a\in\mathbb R^3). This notion guarantees that (F) is non‑negative, but the converse question—whether every non‑negative ternary quartic admits such a representation—has been open for the case of (3\times3) matrices. Buckley and Šivic (2020) conjectured a negative answer and suggested the Robinson polynomial (R) as a possible counterexample.

The authors prove two main theorems. Theorem 1.1 shows that the Robinson polynomial does not admit any positive quadratic determinant representation. The proof proceeds by analyzing the kernel of a putative matrix (M). Lemma 2.1 establishes that if the kernel of (M(a)) has dimension at least two for some complex point (a), then (