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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.15452 |
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| _version_ | 1866910223402270720 |
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| author | Müller, Peter |
| author_facet | Müller, Peter |
| contents | Let $K[x,y,z]=K[X,Y,Z]/(X^2+Y^2+Z^2-1)$ be the coordinate ring of the algebraic unit sphere over a field $K$. Umberto Zannier showed that there exists a matrix in $\operatorname{SL}_3(K[x,y,z])$ with first row $(x,y,z)$ for $K=\mathbb Q_p$, the field of $p$-adic numbers for an odd prime $p$, or more generally, if $-1$ is a sum of two squares in $K$. The case $K=\mathbb Q_2$ remained open and was subsequently posed and discussed by Zannier with numerous researchers, thereby bringing the problem to broader attention.
In 2025, Alexey Ananyevskiy and Marc Levine showed that such a matrix exists if and only if $K$ has Stufe at most $4$, equivalently, if there exist $a,b,c,d\in K$ such that $a^2+b^2+c^2+d^2=-1$. Since $\mathbb Q_2$ has Stufe $4$, this settled Zannier's problem.
Their proof is purely existential and does not provide an explicit matrix. In this note, we construct an explicit example in terms of $a,b,c,d$ and describe the computational techniques used to find it. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_15452 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Explicitly combing hedgehogs over fields of Stufe 4 Müller, Peter Number Theory Algebraic Geometry 14R10 (Primary) 11E25, 57R25 (Secondary) Let $K[x,y,z]=K[X,Y,Z]/(X^2+Y^2+Z^2-1)$ be the coordinate ring of the algebraic unit sphere over a field $K$. Umberto Zannier showed that there exists a matrix in $\operatorname{SL}_3(K[x,y,z])$ with first row $(x,y,z)$ for $K=\mathbb Q_p$, the field of $p$-adic numbers for an odd prime $p$, or more generally, if $-1$ is a sum of two squares in $K$. The case $K=\mathbb Q_2$ remained open and was subsequently posed and discussed by Zannier with numerous researchers, thereby bringing the problem to broader attention. In 2025, Alexey Ananyevskiy and Marc Levine showed that such a matrix exists if and only if $K$ has Stufe at most $4$, equivalently, if there exist $a,b,c,d\in K$ such that $a^2+b^2+c^2+d^2=-1$. Since $\mathbb Q_2$ has Stufe $4$, this settled Zannier's problem. Their proof is purely existential and does not provide an explicit matrix. In this note, we construct an explicit example in terms of $a,b,c,d$ and describe the computational techniques used to find it. |
| title | Explicitly combing hedgehogs over fields of Stufe 4 |
| topic | Number Theory Algebraic Geometry 14R10 (Primary) 11E25, 57R25 (Secondary) |
| url | https://arxiv.org/abs/2605.15452 |