Let ${M}_{d,n}\left(q\right)$ denote the number of monic irreducible polynomials in ${\mathbb{F}}_{q}[{x}_{1},{x}_{2},...,{x}_{n}]$ of degree $d$. We show that for a fixed degree $d$, the sequence ${M}_{d,n}\left(q\right)$ converges coefficientwise to an explicitly determined rational function ${M}_{d,\infty}\left(q\right)$. The limit ${M}_{d,\infty}\left(q\right)$ is related to the classic necklace polynomial ${M}_{d,1}\left(q\right)$ 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.

Revised: 2018-06-20

Accepted: 2018-07-11

Published online: 2019-08-01

DOI: https://doi.org/10.5802/alco.34

Classification: 11T55, 11C08, 11T06

Keywords: necklace polynomial, finite fields, reciprocity

@article{ALCO_2019__2_4_521_0, author = {Hyde, Trevor}, title = {Liminal reciprocity and factorization statistics}, journal = {Algebraic Combinatorics}, pages = {521--539}, publisher = {MathOA foundation}, volume = {2}, number = {4}, year = {2019}, doi = {10.5802/alco.34}, mrnumber = {3997509}, zbl = {07089135}, language = {en}, url = {alco.centre-mersenne.org/item/ALCO_2019__2_4_521_0/} }

Hyde, Trevor. Liminal reciprocity and factorization statistics. Algebraic Combinatorics, Volume 2 (2019) no. 4, pp. 521-539. doi : 10.5802/alco.34. https://alco.centre-mersenne.org/item/ALCO_2019__2_4_521_0/

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