ALGEBRAIC COMBINATORICS

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

Received : 2018-04-27
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}
}

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

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