Xiang Fan

collaborators

10 papers

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…

quant-ph2026

Spectral gaps and slow modes of Pauli rotations and random quantum circuits

Ziyuan Dong, Xiang Fan

We determine which representation sectors control the slowest convergence of random Pauli rotations and several local random quantum circuits. For $d = 2^{n}$ , the continuous Pauli wal…

cs.IT2026

A linear-in- $q$ range of dimensions for the MDS conjecture over $F_{q}$ in odd characteristic

Xiang Fan

Let $q$ be a power of an odd prime $p$ . We prove the MDS conjecture over $F_{q}$ in every dimension $k$ satisfying \[ 2\leqslant k\leqslant B(p,q) \quad\text{or}\quad q+2-B(p…

math.CO2026

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

Xiang Fan

We classify, for every prime power $q$ and every $e ⩾ 2$ , the permutation binomials $X^{r} (X^{q - 1} + a)$ over $F_{q^{e}}$ . Writing $ℓ_{j} (q) = (q^{j} - 1) / (q - 1)$ , such a binomia…

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…

quant-ph2025

Revisiting finite Abelian hidden subgroup problem and its distributed exact quantum algorithm

Ziyuan Dong, Xiang Fan, Tengxun Zhong +1

We revisit the finite Abelian hidden subgroup problem (AHSP) from a mathematical perspective and make the following contributions. First, by employing amplitude amplification, we p…

quant-ph2025

Probabilistic Bounds on the Number of Elements to Generate Finite Nilpotent Groups and Their Applications

Ziyuan Dong, Xiang Fan, Tengxun Zhong +1

This work establishes a new probabilistic bound on the number of elements to generate finite nilpotent groups. Let $φ_{k} (G)$ denote the probability that $k$ random elements generate…

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

Spectral gaps and slow modes of Pauli rotations and random quantum circuits

A linear-in-q range of dimensions for the MDS conjecture over Fq​ in odd characteristic

A complete classification of permutation binomials of the form Xr(Xq−1+a) over finite fields

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

Rational functions over finite fields with Galois closure of genus zero

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

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

Revisiting finite Abelian hidden subgroup problem and its distributed exact quantum algorithm

Probabilistic Bounds on the Number of Elements to Generate Finite Nilpotent Groups and Their Applications

A linear-in- $q$ range of dimensions for the MDS conjecture over $F_{q}$ in odd characteristic

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