Liminal reciprocity and factorization statistics

2018/03/22 by Trevor Hyde, Hyde, Trevor
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #math.CO #math.NT

paper · pdf · doi:10.48550/arxiv.1803.08438

22 pages. To appear in Algebraic Combinatorics

arxiv created 2018/09/06 · arxiv updated 2018/09/10

Abstract

Let M_d,n(q) denote the number of monic irreducible polynomials in \mathbbF_q[x₁, x₂, … , x_n] of degree d. We show that for a fixed degree d, the sequence M_d,n(q) converges q-adically to an explicitly determined rational function M_d,∞(q). Furthermore we show that the limit M_d,∞(q) is related to the classic necklace polynomial M_d,1(q) by an involutive functional equation, leading to a phenomenon we call liminal reciprocity. The limiting first moments of factorization statistics for squarefree polynomials are expressed in terms of a family of symmetric group representations as a consequence of liminal reciprocity.

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