Liminal reciprocity and factorization statistics

Abstract

Let M d,n (q) denote the number of monic irreducible polynomials in Fq[x1, x2, . . . , xn] 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 the twisted Grothendieck-Lefschetz formula of Church, Ellenberg, and Farb.

Stability of the Cohomology of the Space of Complex Irreducible Polynomials in Several Variables

“…We first prove some preliminary results that will be used later in the paper. The results we collect here can be viewed as topological analogs of Lemma 2.1 in [6].…”

“…In 2018, Hyde (Theorem 1.1 in [6]) proved that |Irr d,n (F q )| is always a polynomial in q which converges coefficient-wise to a formal power series P d (q) as n → ∞. In other words, in the formal power series ring Q[[q]] equipped with the q-adic topology (under which higher powers of q are considered smaller), |Irr d,n (F q )| −→ P d (q) as n → ∞.…”

Euler characteristic of the space of real multivariate irreducible polynomials

“…In this section we prove the higher cyclotomic identity. First we recall the definitions of the sets Poly d,n (K), Irr d,n (K) and their associated polynomials P d,n (x), M d,n (x) from [6]. Let d, n ≥ 1 be integers and let K be an arbitrary field.…”

“…In [6] the author introduced the higher necklace polynomials M d,n (x) ∈ Q[x] which interpolate the point counts of Irr d,n (F q ) for finite fields F q where q is a prime power,…”