AI 中文总结
该研究对有限域上拟自同构的有限扩张置换谱分类，反驳相关猜想，给出有限谱映射次数界及特定条件下的谱情况，记录余有限谱准则与素数\(p\)互素问题，还给出非平凡反例几何次数与\(p\)的关系。
AI 中文摘要
我们对有限域上一族明确的拟自同构中出现的有限扩张置换谱进行了分类。\(\N_{>0}\)的每个有限非空因子封闭子集都会出现；特别地，这反驳了Maubach和Willems的猜想，即每个拟自同构都能置换无限多个有限扩张。每个满足\(\Jac(F)=I_n\)且\(F|_{\F_q^n}=\id\)的有限谱映射的总次数至少为\(p(q + 1)\)。对于每个\(n\geq2\)，这个界可由一个谱为\(\{1\}\)的映射达到。在一维情况下，在每个奇数次域和\(\F_2\)上同样的界和精确谱\(\{1\}\)可达到；对于偶数\(q>2\)，我们给出了一个其谱有有效有限界的非例外构造。若\(\delta_p\)表示\(\mathbb F_p\)上非线性约化置换多项式的最小次数，那么非例外单变量拟自同构的最小次数\(\mu_p\)满足\(\mu_2 = 6\)，\(\mu_3 = 12\)，\(\mu_p = p\delta_p\)（\(p\geq5\)）。我们还记录了一个余有限谱准则并分离出素数\(p\)互素的遗留问题。这里构造的每个非平凡反例的几何一般次数都能被\(p\)整除。
英文摘要
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 $p$.
