Algebraic boundaries of Hilbert's SOS cones

TL;DR

This paper explores the geometric and algebraic structures distinguishing non-negative polynomials from sums of squares, focusing on hypersurfaces related to K3 surfaces and their duals, with computational verification of degrees.

Contribution

It identifies the hypersurfaces discriminating non-negative polynomials from SOS as Noether-Lefschetz loci and computes their degrees, linking algebraic geometry with polynomial optimization.

Findings

Hypersurfaces are Noether-Lefschetz loci of K3 surfaces.

Degrees of dual hypersurfaces verified via numerical algebraic geometry.

Non-SOS extreme rays parametrized by Severi varieties and quartic symmetroids.

Abstract

We study the geometry underlying the difference between non-negative 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.