# Two remarks on sums of squares with rational coefficients

## TL;DR

This paper investigates the existence of homogeneous polynomials with rational coefficients that are sums of squares over the reals but not over the rationals, exploring the underlying number field properties and the nature of their zeros.

## Contribution

It relaxes known Galois-theoretic conditions for such polynomials and examines whether these polynomials necessarily have real zeros, providing new insights into their structure.

## Key findings

- Relaxed Galois-theoretic conditions still produce sums of squares over reals but not rationals.
- Examples without real zeros are contained in a thin subset of the boundary of the sum of squares cone.
- Proved that in certain minimal cases, all such examples without real zeros are contained in a thin boundary subset.

## Abstract

There exist homogeneous polynomials $f$ with $\mathbb Q$-coefficients that are sums of squares over $\mathbb R$ but not over $\mathbb Q$. The only systematic construction of such polynomials that is known so far uses as its key ingredient totally imaginary number fields $K/\mathbb Q$ with specific Galois-theoretic properties. We first show that one may relax these properties considerably without losing the conclusion, and that this relaxation is sharp at least in a weak sense. In the second part we discuss the open question whether any $f$ as above necessarily has a (non-trivial) real zero. In the minimal open cases $(3,6)$ and $(4,4)$, we prove that all examples without a real zero are contained in a thin subset of the boundary of the sum of squares cone.