Univariate Rational Sums of Squares

TL;DR

This paper characterizes when a rational polynomial g is non-negative on the real roots of another polynomial f, showing it is equivalent to g being a sum of squares modulo f, and provides an algorithm for certification.

Contribution

It establishes a new theoretical characterization linking non-negativity and sums of squares for rational polynomials on univariate roots, with an accompanying algorithm.

Findings

g is non-negative on roots of f iff g is a sum of squares modulo f

Provided an algorithm to certify non-negativity of g on roots of f

Complete characterization under specific gcd conditions

Abstract

Given rational univariate polynomials f and g such that gcd(f, g) and f / gcd(f, g) are relatively prime, we show that g is non-negative on all the real roots of f if and only if g is a sum of squares of rational polynomials modulo f. We complete our study by exhibiting an algorithm that produces a certificate that a polynomial g is non-negative on the real roots of a non-zero polynomial f , when the above assumption is satisfied.