Prescribed extension spectra of mock automorphisms over finite fields

Abstract

We classify the finite-extension permutation spectra occurring in an explicit family of mock automorphisms over finite fields. Every finite nonempty divisor-closed subset of $\N_{>0}$ occurs; in particular, this disproves Maubach and Willems conjecture that every mock automorphism permutes infinitely many finite extensions. Every finite-spectrum map satisfying \[ \Jac(F)=I_n,\qquad F|_{\F_q^n}=\id \] has total degree at least $p(q+1)$. For every $n\geq2$, this bound is attained by a map with spectrum $\{1\}$. In dimension one, the same bound and exact spectrum $\{1\}$ are attained over every odd field and over $\F_2$; for even $q>2$ we give a nonexceptional construction with an effective finite bound for its spectrum. If $δ_p$ denotes the least degree of a nonlinear reduced permutation polynomial over $\mathbb F_p$, then the least degree $μ_p$ of a nonexceptional one-variable mock automorphism satisfies \[ μ_2=6,\qquad μ_3=12,\qquad μ_p=pδ_p\quad(p\ge5). \] We also record a cofinite-spectrum criterion and isolate the surviving prime-to-$p$ question. Every nontrivial counterexample constructed here has geometric generic degree divisible by $p$.

Disclosure

“te spectrum characterizes polynomial automorphisms among Keller maps. The prime-to-p generic-degree condition is a natural surviving boundary for the constructions considered here and remains unresolved in higher dimension. Declaration of generative AI and AI-assisted technologies in the manuscript preparation process During the preparation of this work, the authors used ChatGPT (GPT-5.6 Sol) for exploratory mathematical brainstorming, manuscript organization and language e”

PDF page 23

Classification

Drafting limited passages

Multiplier

5

Verified

2026-08-23T19:02:36.657514Z

Structural counts