Explicitly combing hedgehogs over fields of Stufe 4

Abstract: Let $K[x,y,z]=K[X,Y,Z]/(X<sup>2+Y<sup>2+Z<sup>2-1)$ 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 $a<sup>2+b<sup>2+c<sup>2+d<sup>2=-1$. 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.

• The paper constructs an explicit matrix in SL₃(K[x,y,z]) with first row (x,y,z) whenever −1 is a sum of at most four squares in K, using entries of degree at most 2 and a constant nonzero determinant.
• The construction resolves the Stufe-4 case, including Q₂, and provides a concrete matrix over Z₂ by using the quadruple (ω/7, 1, 3ω/7, 2ω/7) with ω² = −7.
• The paper connects unimodular completion to nowhere-vanishing tangent vector fields and develops a computational discovery method combining modular lifting, component analysis, interpolation, and sum-of-squares parametrization.

The problem and its history

Let $K[x,y,z] = K[X,Y,Z]/(X^2+Y^2+Z^2-1)$ denote the coordinate ring of the algebraic unit $2$-sphere over a field $K$, with $x,y,z$ the images of $X,Y,Z$. Zannier asked whether there exists a matrix in $\SL_3(K[x,y,z])$ with first row $(x,y,z)$ — a unimodular completion of the row $(x,y,z)$. He answered this affirmatively, in unpublished work, for $K = \mathbb{Q}_p$ with $p$ odd, and posed the case $2$0 to numerous researchers in number theory, algebraic geometry, and algebraic $2$1-theory.

The odd $2$2 case exploits that $2$3 has Stufe $2$4: if $2$5, the explicit matrix with rows $2$6, $2$7, and $2$8 has determinant $2$9. The field $K$0 has Stufe $K$1, so this construction fails, and the case remained open until Ananyevskiy and Levine proved that a positive answer holds if and only if $K$2 has Stufe at most $K$3 — equivalently, $K$4 is a sum of four squares in $K$5. Their proof, however, is purely existential, relying on advanced machinery from algebraic $K$6-theory and motivic homotopy theory, and produces no explicit matrix. The paper under discussion supplies the missing explicit construction.

The main theorem

The central result gives an explicit unimodular matrix over any field of Stufe $K$7, parametrized by a quadruple $K$8 with $K$9. The matrix has first row $x,y,z$0; the second and third rows consist of the polynomials $x,y,z$1, each of degree at most $x,y,z$2 in $x,y,z$3 and polynomial in $x,y,z$4. The determinant is

$x,y,z$5

Since the determinant is constant (independent of $x,y,z$6), the matrix lies in $x,y,z$7 whenever it is nonzero. The paper handles the three Stufe cases separately:

• Stufe 1: take $x,y,z$8, $x,y,z$9; then $X,Y,Z$0.
• Stufe 2: choose $X,Y,Z$1 with $X,Y,Z$2 and $X,Y,Z$3, set $X,Y,Z$4.
• Stufe 4: all of $X,Y,Z$5 and $X,Y,Z$6 are nonzero (otherwise the Stufe would be smaller). If $X,Y,Z$7 for the chosen quadruple, then $X,Y,Z$8; swapping $X,Y,Z$9 and $\SL_3(K[x,y,z])$0 would force $\SL_3(K[x,y,z])$1, a contradiction, so one of the two orderings works.

A notable corollary is a solution over the $\SL_3(K[x,y,z])$2-adic integers $\SL_3(K[x,y,z])$3, a sharpening that the Ananyevskiy–Levine existence result does not address. Taking $\SL_3(K[x,y,z])$4 with $\SL_3(K[x,y,z])$5 yields an explicit matrix over $\SL_3(K[x,y,z])$6 with $\SL_3(K[x,y,z])$7. Since $\SL_3(K[x,y,z])$8, dividing the second row by $\SL_3(K[x,y,z])$9 produces a matrix in $(x,y,z)$0 with determinant $(x,y,z)$1. A similar example was previously communicated by the author to Ananyevskiy and Levine (Remark 1.10 of their paper).

Both theorems are verified by short SageMath scripts included in the paper, which reduce the determinant computations modulo the relation $(x,y,z)$2 (and, for the parametrized version, modulo the constraint $(x,y,z)$3).

Relation to nowhere vanishing vector fields

The matrix formulation is equivalent to the existence of a nowhere vanishing vector field on the algebraic sphere, which explains the title. Given $(x,y,z)$4 with first row $(x,y,z)$5, the second row $(x,y,z)$6 is everywhere linearly independent of the position vector $(x,y,z)$7 on $(x,y,z)$8; projecting out the radial component via

$(x,y,z)$9

yields a tangent vector field that vanishes nowhere on $(x,y,z)$0. Conversely, an appendix gives an elementary version of Ananyevskiy–Levine's Remark 1.7: a nowhere vanishing vector field $(x,y,z)$1 guarantees that the $(x,y,z)$2 minors of $(x,y,z)$3 have no common zero on the sphere, so by Hilbert's Nullstellensatz one can find a third row making the determinant congruent to $(x,y,z)$4 modulo $(x,y,z)$5.

The real case contrasts sharply: by the Hairy Ball Theorem, no such field exists on $(x,y,z)$6, so the answer to Zannier's question is negative for $(x,y,z)$7 — consistent with $(x,y,z)$8 having infinite Stufe. The paper also notes a weaker $(x,y,z)$9-adic variant, requiring nonvanishing only on $K = \mathbb{Q}_p$0 rather than $K = \mathbb{Q}_p$1, which admits a simple solution over $K = \mathbb{Q}_p$2 via $K = \mathbb{Q}_p$3, since $K = \mathbb{Q}_p$4 would force $K = \mathbb{Q}_p$5 to be a square in $K = \mathbb{Q}_p$6.

The search strategy

The paper documents how the parametrized example was found, a nontrivial computational effort combining commutative algebra and computer algebra. Writing each entry of $K = \mathbb{Q}_p$7 as an affine linear polynomial in $K = \mathbb{Q}_p$8 with unknown coefficients $K = \mathbb{Q}_p$9, the condition $p$0 reduces to $p$1 quadratic equations in these $p$2 unknowns, defining a variety $p$3. Symmetry reductions (row/column operations and normalization) cut the problem to $p$4 unknowns with $p$5 and $p$6.

Key steps in the search:

• Reduction mod 2: computing $p$7 by brute force over $p$8 candidates yields $p$9 solutions; a backtracking search shows $2$00 do not lift modulo $2$01, while the remaining four lift to solutions modulo $2$02, strongly suggesting genuine $2$03-points.
• Component detection: specialization experiments suggested $2$04 has at least two components — one of dimension $2$05 with $2$06-points, and one of dimension $2$07 without. Groebner basis computations after fourfold specialization revealed that $2$08 lies in the ideal; since this equation has no solution over a field of Stufe $2$09, adding the slack-variable constraint $2$10 excises the spurious component.
• Interpolation over quadratic fields: direct elimination via Groebner bases exceeded 128 GB of memory even in msolve. Instead, the author computed the $2$11-vector space (dimension $2$12) of polynomials of degree $2$13 in nine of the variables vanishing at thousands of solutions over quadratic extensions of $2$14, then used msolve to eliminate. This revealed that the target variables $2$15 depend only quadratically on $2$16.
• Sum-of-squares parametrization: the lowest-degree relation $2$17 (total degree $2$18) simplifies dramatically under a rational substitution to $2$19, where $2$20 is a sum of five squares of polynomials, $2$21. Since $2$22 and $2$23, $2$24 is positive on $2$25, and the condition $2$26 becomes

$2$27

which is precisely a Stufe-4 representation of $2$28. Matching $2$29 and solving for $2$30 yields explicit rational expressions, from which the full parametrization of Theorem 1 follows.

This last step is conceptually satisfying: the obstruction to combing over $2$31 (positivity of $2$32) is exactly what forces the solution to live over fields admitting a four-square representation of $2$33, mirroring the Ananyevskiy–Levine criterion.

Limitations and open questions

The construction presupposes a Stufe-$2$34 witness $2$35 and, for Stufe $2$36 fields, may require swapping $2$37 and $2$38 to avoid a vanishing determinant; it does not give a canonical matrix. The search heuristic that all entries have degree at most $2$39 was an optimistic assumption, justified only a posteriori by success — the paper does not establish that lower-degree solutions are the only ones, nor characterize the full solution variety $2$40. The claim that the four mod-$2$41 lifts extend to $2$42-points is supported by computation to precision $2$43 rather than by proof, though this suffices for the stated theorems. Two questions remain open: whether the variety $2$44 has exactly the two components inferred from specialization experiments, and whether the weaker variant of the problem (vector fields nonvanishing only on $2$45, rather than $2$46) admits a classification parallel to the Stufe criterion.

Conclusion

This paper complements the existential theorem of Ananyevskiy and Levine with an explicit unimodular completion of $2$47 over the coordinate ring of the algebraic sphere, valid over any field of Stufe at most $2$48, together with a concrete example over $2$49 answering Zannier's sharpened question. Beyond the explicit formulas, the paper's methodological contribution — combining mod-$2$50 lifting, slack-variable component excision, interpolation over quadratic fields, and sum-of-squares parametrization to bypass intractable Groebner computations — offers a reproducible toolkit for similar explicit-construction problems in algebraic $2$51-theory.