For $f(X)=X^r(X^s+a)\in\mathbb F_{p^m}[X]$, with $m,r,s\geqslant1$ and $a\in\mathbb F_{p^m}^\times$, put $g=\gcd(s,p^m-1)$. We prove that, for each fixed $g$, only finitely many prime characteristics $p$ admit such permutation binomials, even as $m,r,s,a$ vary. If $6\nmid g$, every such $p$ divides Wendt's determinant...
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 walk has exact all-representation gaps $$\Delta_{\mathsf{SU}(d)}=\frac{d-8}{8(d-1)}\quad(n\ge4),\qquad \Delta_{\mathsf{PU}(d)}=\frac{d(d-3)}{8(...
We resolve the spectral-gap problem posed by Baer and Haah for random Pauli rotations. In this walk, a nonidentity $n$-qubit Pauli operator $P$ and an angle $\theta$ modulo $2\pi$ are chosen independently and uniformly, and the step is $e^{\mathrm{i}\theta P}$. Writing $d=2^n$, we prove that the gap on $\mathsf{SU}(d)$...
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 $1\leqslant h<e$...
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...
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...
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
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