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

Technical Summary of "Positive Quadratic Determinantal Representations of Quartic Curves and the Robinson Polynomial"

Problem Statement
The paper addresses the existence of positive quadratic determinantal representations for real nonnegative homogeneous polynomials. Specifically, given a homogeneous polynomial $F \in \mathbb{R}[x_0, x_1, x_2]$ of degree $2d$, the authors investigate whether $F$ can be expressed as the determinant of a symmetric $d \times d$ matrix $M$ with entries that are homogeneous quadratic polynomials, such that $M(a)$ is positive semidefinite for all $a \in \mathbb{R}^3$.

While the existence of such representations implies nonnegativity, the converse is a long-standing open question for $d=3$ (degree 6 polynomials). Buckley and Šivic [BŠ20] conjectured that the answer is negative, proposing the Robinson polynomial $R$ (a specific nonnegative sextic) as a counterexample. Furthermore, it was known from Quarez [Qua15] that in degree 4 (quartics), counterexamples exist, but the geometric nature of these counterexamples was not fully characterized.

Methodology
The authors employ a combination of algebraic geometry, sheaf theory, and computational algebra to analyze the structure of determinantal representations.

1. Sheaf-Theoretic Framework: The paper utilizes the correspondence between determinantal representations and coherent sheaves on the curve $X = V(F)$. A quadratic representation corresponds to a line bundle $L$ on $X$ satisfying specific cohomological conditions (specifically related to 2-torsion points in the Picard group).
2. Analysis of Singularities (Robinson Polynomial): For the Robinson polynomial, the authors analyze the singularities of its zero set $X$. They establish that $X$ has ten real singular points and that its normalization is a genus 0 curve with an empty real part. They compute the Picard group $\text{Pic}(X)$, showing it contains a large 2-torsion subgroup isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{10}$.
3. Classification of Representing Types: By explicitly computing the free resolutions of all 1024 2-torsion points in $\text{Pic}(X)$, the authors classify the possible "representing types" (matrices with specific degree patterns). They demonstrate that while determinantal representations exist, none of the types corresponding to purely quadratic entries yield a positive semidefinite matrix.
4. Geometry of Smooth Quartics: For smooth positive ternary quartics ($d=2$), the authors utilize the theory of theta characteristics and Steiner complexes on the Jacobian variety $J(C)$. They relate quadratic representations to contact curves of degree $2d-2$ (conics) and employ a constructive adaptation of Dixon's algorithm.
5. Real Structure and Reality Conditions: The authors distinguish between "real" and "strongly real" divisor classes on the Jacobian. They prove that for a smooth positive quartic, there exist strongly real 2-torsion points that satisfy the necessary cohomological conditions to generate a quadratic representation. They further prove that the resulting matrix is positive definite by analyzing the topological types of the real zero sets of the contact conics involved.

Key Contributions and Results

• Theorem 1.1 (Non-existence for Robinson Polynomial): The authors prove that the Robinson polynomial $R$ does not admit a positive quadratic determinantal representation. This confirms the conjecture by Buckley and Šivic. The proof relies on the fact that any such representation would require a 2-torsion point in $\text{Pic}(X)$ corresponding to a specific line bundle, but an exhaustive analysis of all such bundles reveals that no resulting matrix is positive semidefinite everywhere.
• Theorem 1.2 (Existence for Smooth Quartics): The authors prove that every smooth positive ternary quartic admits a positive quadratic determinantal representation. This result is significant because it establishes that the known counterexamples to the general conjecture (from Quarez) must necessarily define singular plane curves.
• Constructive Algorithm: The paper provides a constructive method (Algorithm 3.8 and Theorem 3.26) to generate a positive quadratic representation for a smooth positive quartic, given a real linear determinantal representation. This involves computing the bitangent matrix, identifying a specific Steiner complex, and applying a change of basis to derive the quadratic matrix $Q$.
• Classification of Representing Types: The paper provides a detailed classification of the representing types for the Robinson polynomial, showing that while quadratic representations exist in a broader sense (with mixed degrees), the strictly quadratic ones fail the positivity condition.

Significance and Claims
The paper claims to resolve the specific question regarding the Robinson polynomial, providing a definitive negative answer to the existence of a positive quadratic representation for this specific counterexample.

Regarding the general case, the paper establishes a sharp geometric boundary: the failure of positive quadratic representations is intrinsically linked to the singularity of the curve. For the class of smooth positive ternary quartics, the authors prove that such representations always exist, thereby characterizing the "good" behavior of this specific degree.

The authors conclude with a modest conjecture (Conjecture 4.1) that the set of nonnegative forms of degree $2d$ admitting a positive quadratic representation is dense in the set of all nonnegative forms of degree $2d$ (with respect to the Euclidean topology). They note that while the set of such forms contains an open neighborhood (demonstrated by perturbing the sum of squares), the extent of this set for higher degrees remains an open question. The paper does not claim to solve the general case for $d \ge 3$ or for singular curves of higher degrees, but rather delineates the precise conditions under which the representation exists for quartics and fails for the specific sextic case.