ALGEBRAIC COMBINATORICS

Liminal reciprocity and factorization statistics
Algebraic Combinatorics, Volume 2 (2019) no. 4, p. 521-539

Let ${M}_{d,n}\left(q\right)$ denote the number of monic irreducible polynomials in ${𝔽}_{q}\left[{x}_{1},{x}_{2},...,{x}_{n}\right]$ 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},
publisher = {MathOA foundation},
volume = {2},
number = {4},
year = {2019},
pages = {521-539},
doi = {10.5802/alco.34},
language = {en},
url = {https://alco.centre-mersenne.org/item/ALCO_2019__2_4_521_0}
}

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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