Sums of squares of polynomials with rational coefficients

TL;DR

This paper constructs explicit polynomials with rational coefficients that are sums of squares over the reals but not over the rationals, addressing an open question and analyzing their representations as sums of squares of rational functions.

Contribution

It provides explicit examples of polynomials that are sums of squares over reals but not over rationals, and characterizes such cases for ternary quartics.

Findings

Constructed explicit polynomials with these properties.

Proved the uniqueness of counterexamples for ternary quartics.

Addressed an open question by Sturmfels.

Abstract

We construct families of explicit polynomials f with rational coefficients that are sums of squares of polynomials over the real numbers, but not over the rational numbers. Whether or not such examples exist was an open question originally raised by Sturmfels. We also study representations of f as sums of squares of rational functions with rational coefficients. In the case of ternary quartics, we prove that our counterexamples to Sturmfels' question are the only ones.