Sharp Degree Bounds for Sum-of-Squares Certificates on Projective Curves

TL;DR

This paper establishes degree bounds for sum-of-squares multipliers on real projective curves, linking algebraic certificates to geometric invariants, and demonstrates the bounds' sharpness through explicit constructions.

Contribution

It introduces geometric invariant-based degree bounds for sum-of-squares multipliers on curves, with new sharpness results and applications to surfaces of minimal degree.

Findings

Degree bounds depend only on geometric invariants of the curve.

Existence of sharp bounds demonstrated via explicit curve deformations.

Generalization of Hilbert's work to surfaces of minimal degree.

Abstract

Given a real projective curve with homogeneous coordinate ring R and a nonnegative homogeneous element f in R, we bound the degree of a nonzero homogeneous sum-of-squares g in R such that the product fg is again a sum of squares. Better yet, our degree bounds only depend on geometric invariants of the curve and we show that there exist smooth curves and nonnegative elements for which our bounds are sharp. We deduce the existence of a multiplier g from a new Bertini Theorem in convex algebraic geometry and prove sharpness by deforming rational Harnack curves on toric surfaces. Our techniques also yield similar bounds for multipliers on surfaces of minimal degree, generalizing Hilbert's work on ternary forms.