Liminal reciprocity and factorization statistics

TL;DR

This paper investigates the asymptotic behavior of the number of irreducible polynomials over finite fields in multiple variables, revealing a new reciprocity phenomenon linking these counts to classical necklace polynomials.

Contribution

It introduces liminal reciprocity, showing the $q$-adic convergence of multivariate irreducible polynomial counts to a rational function related to necklace polynomials.

Findings

$M_{d,n}(q)$ converges $q$-adically to $M_{d, fty}(q)$ for fixed $d$

The limit $M_{d, fty}(q)$ satisfies an involutive functional equation with $M_{d,1}(q)$

First moments of factorization statistics relate to symmetric group representations

Abstract

Let $M_{d,n}(q)$ denote the number of monic irreducible polynomials in $\mathbb{F}_q[x_1, x_2, \ldots , x_n]$ of degree $d$. We show that for a fixed degree $d$, the sequence $M_{d,n}(q)$ converges $q$-adically to an explicitly determined rational function $M_{d,\infty}(q)$. Furthermore we show that the limit $M_{d,\infty}(q)$ is related to the classic necklace polynomial $M_{d,1}(q)$ by an involutive functional equation, leading to a phenomenon we call liminal reciprocity. The limiting first moments of factorization statistics for squarefree polynomials are expressed in terms of a family of symmetric group representations as a consequence of liminal reciprocity.