The Hurwitz sum-of-squares problem depends on the base field

TL;DR

This paper demonstrates that the existence of certain sum-of-squares formulas depends on the base field, providing explicit examples over specific fields and settling a longstanding conjecture.

Contribution

It constructs explicit sum-of-squares formulas over fields where they exist and proves their non-existence over formally real fields, resolving Shapiro's conjecture.

Findings

Explicit formula of type [12,12,18] exists over fields with -1 as a square

No such formula exists over any formally real field

Formula exists over  and , but not over  or 

Abstract

We show that the Hurwitz problem for sums of squares can depend on the base field. More precisely, we construct an explicit formula of type $[12,12,18]$ over every field of characteristic different from $2$ in which $-1$ is a square, whereas no such formula exists over any formally real field. This settles, in the negative, a longstanding conjecture of Shapiro. In particular, a formula of this type exists over $\mathbb Q(i)$ and over $\mathbb C$, but not over $\mathbb Q$ or over $\mathbb R$.