Liminal reciprocity and factorization statistics

Trevor Hyde

doi:10.5802/alco.34

Abstract

Let M d,n (q) denote the number of monic irreducible polynomials in 𝔽 q [x 1 ,x 2 ,...,x n ] of degree d. We show that for a fixed degree d, the sequence M d,n (q) converges coefficientwise to an explicitly determined rational function M d,∞ (q). The limit M d,∞ (q) is related to the classic necklace polynomial M d,1 (q) by an involutive functional equation we call liminal reciprocity. The limiting first moments of factorization statistics for squarefree polynomials are expressed in terms of symmetric group characters as a consequence of liminal reciprocity, giving a liminal analog of a result of Church, Ellenberg, and Farb.

Highlights

Let Fq be a field with q elements
It would be interesting to find a conceptual explanation for this relationship between infinite and one dimensional factorization statistics
A factorization statistic is a function P defined on Polyd,n(Fq) such that P (f ) only depends on the factorization type of f ∈ Polyd,n(Fq)

Summary

Introduction

Let Fq be a field with q elements. How many degree d polynomials in Fq[x1, x2, . Let Md,n(q) denote the number of irreducible monic(1) polynomials in Fq[x1, x2, . When n = 1, Md,1(q) is given by the dth necklace polynomial (1). Md,n(q) analogous to (1) when n > 1. In Lemma 2.1 Md,n(q) is shown to be a recursively computable polynomial in q for all n 1. Manuscript received 27th April 2018, revised 20th June 2018 and 11th July 2018, accepted 11th July 2018.

Polynomial factorization statistics

Liminal first moments of squarefree factorization statistics