The Hurwitz sum-of-squares problem depends on the base field
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909007977906176 |
|---|---|
| author | Zhang, Chi Zhu, Haoran |
| author_facet | Zhang, Chi Zhu, Haoran |
| contents | 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 $\mathbb R$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_00590 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Hurwitz sum-of-squares problem depends on the base field Zhang, Chi Zhu, Haoran Number Theory Algebraic Geometry Combinatorics 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 $\mathbb R$. |
| title | The Hurwitz sum-of-squares problem depends on the base field |
| topic | Number Theory Algebraic Geometry Combinatorics |
| url | https://arxiv.org/abs/2605.00590 |