Algebraic boundaries of Hilbert’s SOS cones

Abstract

Abstract. We study the geometry underlying the difference between nonnegative polynomials and sums of squares. The hypersurfaces that discriminate these two cones for ternary sextics and quaternary quartics are shown to be Noether-Lefschetz loci of K3 surfaces. The projective duals of these hypersurfaces are defined by rank constraints on Hankel matrices. We compute their degrees using numerical algebraic geometry, thereby verifying results due to Maulik and Pandharipande. The non-SOS extreme rays of the two cones of non-negative forms are parametrized respectively by the Severi variety of plane rational sextics and by the variety of quartic symmetroids.

Enumeration of surfaces containing an elliptic quartic curve

“…They are also grateful to Kristian Ranestad and Rahul Pandharipande for correcting an error in §3.2 and for calling their attention to [3] and [10].…”

New examples of extremal positive linear maps

“…By analogy with Blekherman's geometric explanation [2] of the containment of the convex cone Σ 3,6 of SOS polynomials in the convex cone P 3,6 ⊂ P 27 of nonnegative ternary polynomials of degree 6, we can study the containment of the convex cone Σ B (3,3) of SOS biquadratic forms in the convex cone P B (3,3) of nonnegative biquadratic forms on P 2 × P 2 , or equivalently the containment of the convex cone of completely positive maps in the convex cone of positive maps from Sym 3 to Sym 3 . Determinant is a natural map det : P B(3,3) → P 3,6 p Φ (x, y) → det Φ(x x T ) that is obviously not injective.…”

“…has no semidefinite quadratic determinantal representation. It is not hard to see that det maps Σ B (3,3) into Σ 3,6 , since for a completely positive map Φ, the determinant det Φ(x x T ) is an SOS polynomial [30]. On the other hand, Quarez' example [30,Proposition 5.1] with determinant det  …”

“…Our explicit examples provide a big family of (extremal) positive maps that might help to understand the difference between the convex cones Σ B(3,3) ⊂ P B (3,3) and moreover their boundaries. Repeating the proofs of [26] and [3] for biquadratic forms, shows that the algebraic boundary of P B (3,3) is the discriminant of biquadratic ternary forms. Using [14,Chapter 13.…”

Existence and Density of General Components of the Noether–Lefschetz Locus on Normal Threefolds

“…Even though both papers obtain density results by using determinantal curves, there are substantial differences in both the results and the methods. Benoist's paper, as well as [26] and [5] use the density results for Noether-Lefschetz in smooth loci. The current paper opens the way to study such problems in more general contexts.…”