Explicitly combing hedgehogs over fields of Stufe 4
Pith reviewed 2026-05-19 14:24 UTC · model grok-4.3
The pith
Explicit formulas produce a matrix in SL_3 over the unit sphere ring from four squares summing to minus one.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We construct an explicit example in terms of a, b, c, d of a matrix in SL_3(K[x, y, z]) with first row (x, y, z) whenever a squared plus b squared plus c squared plus d squared equals minus one in K.
What carries the argument
The explicit 3-by-3 matrix whose nine entries are polynomials in a, b, c, d, x, y, z chosen so that the determinant identity holds after reduction modulo x squared plus y squared plus z squared minus one.
If this is right
- An explicit matrix now exists for every field of Stufe four, including the 2-adics.
- The same technique yields a concrete way to complete the vector (x, y, z) to an invertible matrix over the sphere ring.
- The computational search method used to locate the entries can be repeated for related problems in higher matrix sizes or different quadratic relations.
Where Pith is reading between the lines
- The explicit entries may expose a hidden relation between four-square identities and the geometry of the three-dimensional sphere.
- Having the formulas in hand makes it feasible to test further algebraic properties, such as whether the matrix can be chosen to satisfy additional symmetry or positivity conditions.
Load-bearing premise
The chosen expressions for the matrix entries satisfy the determinant identity when substituted into the quotient ring where x squared plus y squared plus z squared equals one.
What would settle it
Pick concrete a, b, c, d in the 2-adics with a squared plus b squared plus c squared plus d squared equal to minus one, substitute the given formulas into the determinant, and check whether the result reduces to exactly one inside the quotient ring.
read the original 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper constructs an explicit 3×3 matrix with entries in the coordinate ring K[x,y,z] = K[X,Y,Z]/(X² + Y² + Z² - 1) whose first row is (x, y, z) and whose determinant is 1, whenever the base field K admits elements a, b, c, d satisfying a² + b² + c² + d² = -1. The construction is given directly in terms of these four elements and is motivated by the recent existence theorem of Ananyevskiy–Levine that such a matrix exists precisely when the Stufe of K is at most 4.
Significance. The explicit formulas supplied here convert the existential result into a concrete, computable matrix. This is a clear advance for fields of Stufe 4 (including ℚ₂) and supplies a concrete object that can be used for further algebraic or arithmetic investigations. The authors also document the computer-algebra techniques employed to locate the expressions, which adds reproducibility value to the note.
minor comments (3)
- The verification that the constructed matrix has determinant 1 is stated to follow from polynomial identities under the two given relations, but the key simplification steps are not displayed. Including at least the leading terms of the determinant expansion (or a brief outline of the computer-assisted check) would make the central claim easier to inspect without external software.
- Notation for the coordinate ring is introduced in the abstract but the precise ideal of relations is restated only informally in the introduction. A single displayed equation defining R = K[X,Y,Z]/(X²+Y²+Z²-1) early in §1 would improve readability.
- The title uses the informal phrase “combing hedgehogs”; a short footnote explaining the origin of the metaphor (or its relation to the topological or algebraic problem) would help readers outside the immediate circle of specialists.
Simulated Author's Rebuttal
We thank the referee for the positive assessment of our manuscript and for recommending minor revision. The referee's summary accurately captures the main contribution: an explicit matrix in SL_3(K[x,y,z]) with first row (x,y,z) whenever a,b,c,d exist with a² + b² + c² + d² = -1. We appreciate the recognition that this converts the Ananyevskiy–Levine existence result into a concrete, computable object and that documenting the computer-algebra search adds reproducibility.
Circularity Check
No significant circularity
full rationale
The paper constructs explicit matrix entries as polynomials in a,b,c,d (with the single external relation a²+b²+c²+d²=-1) and verifies the required properties by direct substitution into the quotient ring K[x,y,z]/(x²+y²+z²-1). These verifications are ordinary polynomial identities that hold independently of the target matrix; they do not define the matrix in terms of itself, rename a fitted quantity, or rest on a self-citation chain whose content is the result being proved. The prior existence theorem of Ananyevskiy-Levine is cited only for context and is not invoked to justify the explicit formulas or their correctness.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption K contains elements a,b,c,d satisfying a² + b² + c² + d² = -1
Reference graph
Works this paper leans on
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[1]
Combing a hedgehog over a field
A. Ananyevskiy and M. Levine. “Combing a hedgehog over a field”. In: Algebra Number Theory19.12 (2025), pp. 2409–2431.doi:10.2140/ant. 2025.19.2409
work page doi:10.2140/ant 2025
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[2]
msolve: A Library for Solv- ing Polynomial Systems
J. Berthomieu, C. Eder, and M. Safey El Din. “msolve: A Library for Solv- ing Polynomial Systems”. In:2021 International Symposium on Symbolic and Algebraic Computation. 46th International Symposium on Symbolic and Algebraic Computation. Saint Petersburg, Russia: ACM, July 2021, pp. 51–58.doi:10.1145/3452143.3465545
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[3]
Müller.SageMath verification script.https://ypfmde.github.io/ verify_combing.html
P. Müller.SageMath verification script.https://ypfmde.github.io/ verify_combing.html. 2026
work page 2026
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[4]
The Sage Developers.SageMath, the Sage Mathematics Software System. https://www.sagemath.org. 2022. Institute of Mathematics, University of Würzburg Email address:peter.mueller@uni-wuerzburg.de
work page 2022
discussion (0)
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