Xiang Fan

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math.NT2026

Wendt's determinant and finiteness of prime characteristics for permutation binomials

Zhichao Tang, Xiang Fan

For $f (X) = X^{r} (X^{s} + a) \in F_{p^{m}} [X]$ , with $m, r, s ⩾ 1$ and $a \in F_{p^{m}}^{\times}$ , put $g = g cd (s, p^{m} - 1)$ . We prove that, for each fixed $g$ , only finitely many pr…

math.NT2026

Indecomposable rational functions over finite fields with Galois closure of genus one: Reconstruction and exceptionality

Xiang Fan

Let $k = F_{q}$ . We reconstruct every $k$ -indecomposable $g \in k (X)$ of degree greater than one whose normal closure has genus one. Such a map is automatically separable and, a…

math.NT2026

Rational functions over finite fields with Galois closure of genus zero

Xiang Fan

Let $k = F_{q}$ . 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 t…

math.NT2026

Exceptional covers of the projective line with genus-one Galois closure: arithmetic forms over finite fields

Xiang Fan

Let $k = F_{q}$ . We classify, up to two-sided $k$ -Möbius equivalence, all separable indecomposable tame exceptional maps $P_{k}^{1} \to P_{k}^{1}$ whose geometric Galois…

math.NT2026

Shifted dihedral isogeny quotients and the classification of exceptional rational functions of degree five

Zhichao Tang, Xiang Fan

We give a complete classification, up to $k$ -Möbius equivalence, of exceptional rational functions of degree five over every finite field. The classification gives explicit normal…