We study the rational canonical form of sparse random matrices over a finite field. Suppose $A_n\in \operatorname{Mat}_n(\mathbb{F}_p)$ has independent and $\alpha_n$-balanced entries. We prove that if $$ \liminf_{n\to\infty}\frac{n\alpha_n}{\log n}>1, $$ then, for every fixed collection of distinct monic irreducible polynomials over $\mathbb{F}_p$, the corresponding primary partitions of $A_n$ converge jointly to the same asymptotically independent Cohen-Lenstra distributions as in the uniform model studied by Fulman in his thesis. The sharp sparsity threshold for the full rational canonical form coincides with the threshold previously obtained by Lee for finite-field cokernels and by Jung-Lee-Yu for random matrix models over $\mathbb{Z}_p$. Our proof is based on the surjection moment method over the function field $\mathbb{F}_p[t]$, applied to the finite module $\operatorname{Cok}_{\mathbb{F}_p[t]}(tI_n-A_n)$, whose primary decomposition records the rational canonical form. We also construct degree-$d$ critical sparse obstructions, suggesting a polynomial-dependent threshold $1/d$ for statistics associated with irreducible polynomials of minimal degree $d$.
We investigate the equal-characteristic case of the Moon--Taguchi bounded-conductor finiteness conjecture for mod $p$ Galois representations over global function fields. We first establish a conditional finiteness theorem: for any global function field $K$ of characteristic $p$ and any integer $n \ge 1$, there are only...
For a countable family of hyperplanes $H_j\subset \mathbb{P}^m$, $j\in\mathbb{N}$, in general position and a linearly nondegenerate holomorphic curve $f\colon \mathbb{C}\to \mathbb{P}^m$ of finite lower order, we prove that the Nevanlinna defects $\delta_f(H_j)$ satisfy $$ \sum_{j=1}^{\infty}\delta_f(H_j)^{1/3}<\infty....
Let $f(t)\in\mathbb{Z}[t]$ be a nonconstant polynomial with nonzero discriminant and let $k\ge2$ be an integer. Inspired by the work of Alon and Bourgain, for sufficiently large prime $p\equiv1\pmod{k}$, we study cyclic orderings of subsets $A\subseteq \mathbb{F}_p$ for which $ f(a_i+a_{i+1})$ is a nonzero $k$-th power...
Let $k=\mathbb{F}_q$. A polynomial representative of a finite-set map is a tuple of polynomials inducing that map on the rational-point grid. We prove a sharp distinction between a finite-set map and the geometry of its representatives. If $n=1$ or $n=2$, every polynomial representative of a permutation of $k^n$ has al...
Finite free convolution is obtained by averaging characteristic polynomials over Haar unitary conjugation. We ask when two maximal abelian $\ast$-subalgebras of the complex matrix algebra ${\sf M}_n$ can be placed in finite free position: that is, when their relative position realizes this averaging exactly for every p...
For fixed $k\ge3$, we determine the critical law of totally isotropic $r$-spaces for a uniform random map $\Theta_N:\Lambda^k\mathbb {F}_q^N\to\mathbb {F}_q^m$. At the exact balance $m\binom rk=r(N-r)$, the entire null configuration is asymptotically an independent Bernoulli subset of $\operatorname{Gr}(r,\mathbb {F}_q...
Pakin Methawisal· 0 citations
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