发表机构
School of Mathematics, Sun Yat-sen University（中山大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文分类了有限域上伽罗瓦闭包亏格零的可分有理函数，并证明此类函数置换有限域当且仅当例外，给出置换扩张次数与多项式代表元。
AI 中文摘要
设 $k=\mathbb F_q$。我们在 $k$-Möbius 变换的前后复合意义下，分类所有次数大于 1 的可分 $k$-不可分解有理函数 $f\in k(X)$，其伽罗瓦闭包具有亏格零。该分类在任意特征下有效，并包含精确的算术条件和在给定域上的类计数。半线性 Frobenius 下降确定了有限域形式以及哪些几何分解下降到 $k$。对于每个次数大于 1 且伽罗瓦闭包亏格为零的可分 $f\in k(X)$，以及每个 $m\geqslant1$，我们证明 $f$ 置换 $\mathbf P^1(\mathbb F_{q^m})$ 当且仅当它在 $\mathbb F_{q^m}$ 上是例外的，即它置换无穷多个有限扩张 $L/\mathbb F_{q^m}$ 上的 $\mathbf P^1(L)$。不需要不可分解性假设或对 $q$ 的下界。论证结合了不动点平均与伽罗瓦闭包曲线上的分歧。若全常数域为 $\mathbb F_{q^d}$，这些性质仅依赖于 $\gcd(m,d)$。对于每个分类族，我们明确确定置换扩张次数，并刻画多项式代表元。
英文摘要
Let $k=\mathbb F_q$. We classify, up to pre- and post-composition by $k$-Möbius transformations, all separable $k$-indecomposable rational functions $f\in k(X)$ of degree greater than one whose Galois closure has genus zero. The classification is valid in arbitrary characteristic and includes exact arithmetic conditions and class counts over the prescribed field. Semilinear Frobenius descent determines the finite-field forms and which geometric decompositions descend to $k$. For every separable $f\in k(X)$ of degree greater than one with Galois closure of genus zero and every $m\geqslant1$, we prove that $f$ permutes $\mathbf P^1(\mathbb F_{q^m})$ if and only if it is exceptional over $\mathbb F_{q^m}$, meaning that it permutes $\mathbf P^1(L)$ for infinitely many finite extensions $L/\mathbb F_{q^m}$. No indecomposability assumption or lower bound on $q$ is needed. The argument combines fixed-point averaging with ramification on the Galois-closure curve. If the full constant field is $\mathbb F_{q^d}$, these properties depend only on $\gcd(m,d)$. For each classified family we determine the permutation extension degrees explicitly and characterize polynomial representatives.
Comments81 pages
