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Liminal reciprocity and factorization statistics. (English) Zbl 07089135
Summary: Let \(M_{d,n}(q)\) denote the number of monic irreducible polynomials in \(\mathbb{F}_q[x_1, x_2, \ldots , x_n]\) 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 \(M_{d,\infty }(q)\). The limit \(M_{d,\infty }(q)\) 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.
MSC:
11T55 Arithmetic theory of polynomial rings over finite fields
11C08 Polynomials in number theory
11T06 Polynomials over finite fields
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References:
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