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

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.

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DOI: 10.5802/alco.34
Classification: 11T55, 11C08, 11T06
Keywords: necklace polynomial, finite fields, reciprocity
Hyde, Trevor 1

1 University of Michigan Dept. of mathematics 530 Church St. Ann Arbor MI 48109 USA
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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/articles/10.5802/alco.34/

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