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arXiv · 2605.00590

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

Chi Zhang·Haoran Zhu
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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. In particular, a formula of this type exists over $\mathbb Q(i)$ and over $\mathbb C$, but not over $\mathbb Q$ or over $\mathbb R$. This settles, in the negative, a longstanding conjecture of Shapiro from 1984, a conjecture of Adem from 1975, and answers a signed-formula problem raised by Shapiro in 2000.

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Chi Zhang, Haoran Zhu. 2026-06-11. The Hurwitz sum-of-squares problem depends on the base field. https://arxiv.org/abs/2605.00590

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