Algebraic boundaries of Hilbert's SOS cones

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.

18 pages

References in corpus (1)

• Computing Linear Matrix Representations of Helton-Vinnikov Curves

Cited by in corpus (6)

• Sums of three squares and Noether-Lefschetz loci
• Topological Noetherianity of polynomial functors II: base rings with Noetherian spectrum
• Real Zeros of SONC Polynomials
• The Noether-Lefschetz locus of surfaces in ${\mathbb{P}}^3$ formed by determinantal surfaces
• New examples of extremal positive linear maps
• Quantum Entanglement, Symmetric Nonnegative Quadratic Polynomials and Moment Problems