Extreme Rays of the Hankel Spectrahedra for Ternary Forms

TL;DR

This paper investigates the structure of Hankel spectrahedra for ternary forms, revealing their extreme rays form an irreducible variety and providing explicit constructions and rank stratifications.

Contribution

It characterizes the algebraic boundary of sums of squares cones and explicitly constructs extreme rays using algebraic geometry techniques.

Findings

The Zariski closure of all extreme rays forms an irreducible variety of codimension 10.

Explicit rational extreme rays of maximal rank are constructed.

Rank stratification of extreme rays is detailed for degrees 3, 4, and 5.

Abstract

Hankel spectrahedra are the dual convex cones to the cone of sums of squares of real polynomials, and we study them from the point of view of convex algebraic geometry. We show that the Zariski closure of the union of all extreme rays of Hankel spectrahedra for ternary forms is an irreducible variety of codimension 10. It is the variety of all Hankel (or middle Catalecticant) matrices of corank at least 4. We explicitly construct a rational extreme ray of maximal rank using the Cayley-Bacharach Theorem for plane curves. We work out the rank stratification of the semi-algebraic set of extreme rays of Hankel spectrahedra in the first three nontrivial cases d = 3, 4, 5. Dually, we get a characterisation of the algebraic boundary of the cone of sums of squares via projective duality theory, extending previous work of Blekherman, Hauenstein, Ottem, Ranestad, and Sturmfels.