Nonnegative polynomials from vector bundles on real curves

TL;DR

This paper investigates the nonnegativity of the E-resultant for certain vector bundles on real curves and demonstrates conditions under which it cannot be expressed as a sum of squares, revealing new algebraic properties.

Contribution

It introduces new nonnegativity results for the E-resultant of rank 2 vector bundles on real curves and identifies conditions preventing sum of squares representations.

Findings

E-resultant is nonnegative on real sections for certain bundles.

Under specific degree conditions, the polynomial cannot be a sum of squares.

Provides algebraic criteria linking vector bundle properties to polynomial representations.

Abstract

We observe that the E-resultant of a very ample rank 2 vector bundle E on a real projective curve (with no real points) is nonnegative when restricted to the space of real sections. Moreover, we show that if E has a section vanishing at exactly two points and the degree d of E satisfies d(d-6)> 4g-5, then this polynomial cannot be written as a sum of squares.