Full wreath-product monodromy through the third iterate of an explicit Keller map

Abstract

Let \(F:\mathbb C^3\to\mathbb C^3\) be the announced noninjective polynomial map with constant Jacobian determinant \(-2\) and geometric monodromy \(S_3\). This note provisionally determines the geometric monodromy of its second and third iterates as the full imprimitive wreath products \(W_2=S_3\wr S_3\) and \(W_3=S_3\wr S_3\wr S_3\). It also proves that every iterate \(F^{\circ m}\) has a full \(3^m\)-cycle in its geometric inertia at infinity.

1. The map and theorem

With (u=1+xy), define

Exact expansion gives (det JF=-2), and the three distinct rational points ((0,0,-1/4)), ((1,-3/2,13/2)), and ((-1,3/2,13/2)) have the same image.

2. Exact eliminant and function-field tower

On the target line ((1,2,s)), two inverse-resolvent cubics eliminate to a primitive degree-nine polynomial (P(r,s)). The strengthened edition does more than record the resultant: exact denominator resultants show that the rational reconstruction is defined generically, and a linear subresultant (lambda(r,s)t+mu(r,s)) has (gcd(P,lambda)=1). Therefore (t=-mu/\lambda) is recovered at the generic point and

This rules out an extraneous generic branch introduced by clearing denominators or taking the resultant.

3. Three inertia elements force the wreath product

The Newton polygon of (P) at (s=\infty) has one edge from ((0,0)) to ((9,7)), producing a 9-cycle. Its exact discriminant factors as

where the degree-12 factor (A) is squarefree and coprime to the other displayed factors and the leading coefficient. A loop around a root of (A) gives one within-block transposition. The outer cubic has discriminant (-4q(s)), producing a transposition of the three blocks. A short group lemma then forces all of (S_3^3\rtimes S_3).

4. Full-cycle inertia for all iterates

Over (K=\mathbb C((1/s))), the revised proof tracks an inverse tower in a Puiseux closure. At level (k), the cubic coefficient valuations are ((-c_k,0,-a_k,a_k)). The strict inequality (c_k>5a_k) gives a single endpoint Newton edge, while explicit dominance inequalities rule out cancellation in every reconstruction formula. The recurrence

makes the new parameter valuation have exact denominator (3^{k+1}). Each cubic step is therefore totally ramified of degree three. The level-(m) branch exhausts the (3^m)-sheet slice and tame inertia acts as a single (3^m)-cycle. This does not prove the full iterated wreath product at every level.

5. The third-iterate certificate

Three inverse-resolvent cubics eliminate to a primitive polynomial \(Q(q,s)\) of degree 27 in \(q\). Its normalized lower Newton polygon is the single edge from \((0,0)\) to \((27,34)\), giving irreducibility and a 27-cycle. Exact PARI arithmetic factors the degree-1612 discriminant by multiplicity and isolates a squarefree degree-76 divisor that occurs exactly once and is coprime to the leading coefficient and both reconstruction-denominator norms.

Local inertia at a root of that divisor is a single transposition in one bottom block. The proved \(W_2\) quotient, the 27-cycle, and this transposition generate the full kernel \(S_3^9\), hence all of \(W_3\). The verifier recomputes the eliminant invariants and checks the group lemma without enumerating the more than ten million elements of the kernel.

6. Unit-Jacobian normalization

Since (det J(F\circ F)=4), postcomposing by (operatorname{diag}(1/4,1,1)) gives a noninjective polynomial self-map of (mathbb C^3) with Jacobian determinant one and the same degree-nine wreath-product monodromy.

7. Exact certificates

• verify_symbolic.py — Jacobian, collisions, resolvent reconstruction, exact resultant, denominator control, linear subresultant, Newton edge, discriminant, squarefreeness, and coprimality.
• verify_modular.py — dependency-free finite-field arithmetic and three Frobenius cycle types.
• verify_iterate_inertia.py — exact recurrence, Newton-edge inequalities, and all reconstruction dominance inequalities.
• verify_pari.gp — independent PARI/GP discriminant and arithmetic Galois checks.
• verify_group.g — exhaustive GAP subgroup exclusion inside (S_3\wr S_3).
• verify_level3_newton.py — independent degree-27 Newton edge for the third iterate.
• verify_level3_wreath.py — primitive degree-27 eliminant, multiplicity-one discriminant divisor, denominator guards, and \(W_3\) kernel lemma.
• w4_search/RESULT.md — separate bounded-memory proof of \(W_4\), including good reduction, norm-to-inertia, a direct sheet derivative, and the \(S_3^{27}\) kernel lemma.

8. Scope and priority

The closest public weighted-lift monodromy note states only the containment \(\operatorname{Mon}(F\circ F)\le S_3\wr S_3\) and calls the exact group a natural next computation. A timestamped audit of 23 repositories, arXiv, MathOverflow, and web sources found no earlier exact determination of \(W_2\) or \(W_3\) for this map. That is source-specific evidence, not a guarantee of worldwide priority.

A separate four-variable construction already claims a different imprimitive degree-eight group of order 192. This note therefore does not claim the first imprimitive or first non-symmetric Keller monodromy. Its narrow candidate contribution is the exact second- and third-iterate groups, the pairs \((9,W_2)\) and \((27,W_3)\), and the all-iterate full-cycle statement. The subsequently certified \(W_4\) computation is intentionally kept as a separate proof artifact rather than folded into this paper.

AI-assistance and verification disclosure

The exploratory algebra, source search, proof drafting, verification programs, and website were developed with heavy assistance from ChatGPT 5.6 Sol under Alec Kriebel's direction. Alec Kriebel is a complete amateur and cannot independently verify the claims. The note has not been peer reviewed.

Suggested citation

Alec Kriebel, with heavy assistance from ChatGPT 5.6 Sol, “Full wreath-product monodromy through the third iterate of an explicit Keller map,” provisional research note, first repository release 21 July 2026, third-iterate upgrade 22 July 2026. Permanent page.