Explicitly combing hedgehogs over fields of Stufe 4

TL;DR

This paper constructs an explicit example of a matrix over the coordinate ring of the sphere in fields of Stufe 4, addressing a problem posed by Zannier and building on recent theoretical results.

Contribution

It provides an explicit construction of the matrix, complementing previous existential proofs and detailing the computational methods used.

Findings

Explicit matrix constructed over the coordinate ring of the sphere.

Demonstrates computational techniques for finding such matrices.

Addresses Zannier's open problem for the case of , with Stufe 4 fields.

Abstract

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.