Rational functions over finite fields with Galois closure of genus zero
Abstract
Let k=𝔽q. We classify, up to pre- and post-composition by k-Möbius transformations, all separable k-indecomposable rational functions f∈ 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∈ k(X) of degree greater than one with Galois closure of genus zero and every mgeqslant1, we prove that f permutes P1(𝔽qm) if and only if it is exceptional over 𝔽qm, meaning that it permutes P1(L) for infinitely many finite extensions L/𝔽qm. 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 𝔽qd, these properties depend only on gcd(m,d). For each classified family we determine the permutation extension degrees explicitly and characterize polynomial representatives.
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BibTeX
@misc{fan2026rational,
title = {{Rational functions over finite fields with Galois closure of genus zero}},
author = {Xiang Fan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08857},
eprint = {2609.08857},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08857},
}
RIS
TY - UNPB
TI - Rational functions over finite fields with Galois closure of genus zero
AU - Xiang Fan
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08857
UR - https://arxiv.org/abs/2609.08857
AB - Let k=𝔽_q. We classify, up to pre- and post-composition by k-Möbius transformations, all separable k-indecomposable rational functions f∈ 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∈ k(X) of degree greater than one with Galois closure of genus zero and every mgeqslant1, we prove that f permutes P^1(𝔽_q^m) if and only if it is exceptional over 𝔽_q^m, meaning that it permutes P^1(L) for infinitely many finite extensions L/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 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.
ER -