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Xiang Fan

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Preprint Sep 2026

A complete classification of permutation binomials of the form $X^r(X^{q-1}+a)$ over finite fields

We classify, for every prime power $q$ and every $e\geqslant2$, the permutation binomials $X^r(X^{q-1}+a)$ over $\mathbb F_{q^e}$. Writing $\ell_j(q)=(q^j-1)/(q-1)$, such a binomial is a permutation if and only if $\gcd(r,q-1)=1$, $(-a)^{\ell_e(q)}\ne1$, and $r\ell_h(q)\equiv1\pmod{\ell_e(q)}$ for some $1\leqslant h<e$...

Xiang Fan · 1 citation

Preprint Aug 2026

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

Let $k=\mathbb F_q$. 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, after independent degree-one changes of the source and target coordinates over $k$, arises from a separable equivariant isogeny between ellip...

Xiang Fan · 0 citations

Preprint Sep 2026

Rational functions over finite fields with Galois closure of genus zero

Let $k=\mathbb F_q$. We classify, up to pre- and post-composition by $k$-M\"obius 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 conditi...

Xiang Fan · 1 citation · ⚡1

Preprint Aug 2026

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

Let $k=\mathbb F_q$. We classify, up to two-sided $k$-M\"obius equivalence, all separable indecomposable tame exceptional maps $\mathbf P^1_k\to\mathbf P^1_k$ whose geometric Galois closure has genus one. The resulting arithmetic fixed-field forms are encoded by Frobenius-stable affine elliptic quotient data recovered...

Xiang Fan · 1 citation