Nonnegativity certificates on real algebraic surfaces

TL;DR

This paper develops methods to transfer nonnegativity certificates on real algebraic surfaces, leading to improved bounds and characterizations for sum-of-squares representations of nonnegative forms.

Contribution

It introduces tools for transferring nonnegativity certificates and applies them to improve bounds and characterize nonnegative forms on specific algebraic surfaces.

Findings

Improved Hilbert's degree bounds for sum-of-squares multipliers on ternary forms

Complete characterization of nonnegative forms on del Pezzo surfaces

Quadratic upper bounds for sum-of-squares multipliers on real ruled surfaces

Abstract

We introduce tools for transferring nonnegativity certificates for global sections between line bundles on real algebraic surfaces. As applications, we improve Hilbert's degree bounds on sum-of-squares multipliers for nonnegative ternary forms, give a complete characterization of nonnegative real forms of del Pezzo surfaces, and establish quadratic upper bounds for the degrees of sum-of-squares multipliers for nonnegative forms on real ruled surfaces.