Let denote the number of monic irreducible polynomials in of degree . We show that for a fixed degree , the sequence converges coefficientwise to an explicitly determined rational function . The limit is related to the classic necklace polynomial 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 = {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. https://alco.centre-mersenne.org/item/ALCO_2019__2_4_521_0/
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