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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
##### Keywords:
necklace polynomial; finite fields; reciprocity
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##### References:
 [1] Bodin, A., Number of irreducible polynomials in several variables over finite fields, Am. Math. Mon., 115, 653-660, (2008) · Zbl 1439.11001 [2] Carlitz, L., The arithmetic of polynomials in a Galois field, Proc. Natl. Acad. Sci. U.S.A., 17, 120-122, (1931) · Zbl 1219.12003 [3] Carlitz, L., The arithmetic of polynomials in a Galois field, Am. J. Math., 54, 39-50, (1932) · JFM 57.0171.01 [4] Carlitz, L., The distribution of irreducible polynomials in several indeterminates, Illinois J. Math., 7, 371-375, (1963) · JFM 58.0150.01 [5] Carlitz, L., The distribution of irreducible polynomials in several indeterminates II, Canad. J. Math., 17, 261-266, (1965) · Zbl 0118.26002 [6] Church, T.; Ellenberg, J.; Farb, B., Algebraic Topology: Applications and New Directions, 620, Representation stability in cohomology and asymptotics for families of varieties over finite fields, 1-54, (2014), American Mathematical Society · Zbl 0135.01704 [7] Cohen, S. D., The distribution of irreducible polynomials in several indeterminates over a finite field, P. Edinburgh Math. Soc., 16, 1-17, (1968) · Zbl 1388.14148 [8] Hou, X.-D.; Mullen, G., Number of irreducible polynomials and pairs of relatively prime polynomials in several variables over finite fields, Finite Fields Appl., 15, 304-331, (2009) · Zbl 0172.05305 [9] Hyde, T., Cyclotomic factors of necklace polynomials, (2018) · Zbl 1190.11063 [10] Hyde, T., Polynomial factorization statistics and point configurations in $$\mathbb{R}^3$$, Int. Math. Res. Not., (2018) [11] Hyde, T.; Lagarias, J. C., Polynomial splitting measures and cohomology of the pure braid group, Arnold. Math. J., 3, 219-249, (2017) · Zbl 06855337 [12] Metropolis, N.; Rota, G.-C., Witt vectors and the algebra of necklaces, Adv. Math., 50, 95-125, (1983) · Zbl 06855337 [13] Rosen, M., Number theory in function fields, 120, (2013), Springer Science & Business Media: Springer Science & Business Media, New York · Zbl 0545.05009 [14] Sloane, N. J. A., The On-Line Encyclopedia of Integer Sequences, (2018) · Zbl 1439.11001 [15] Stanley, R. P., Combinatorial reciprocity theorems, Adv. Math., 14, 194-253, (1974) · Zbl 0299.05008 [16] Von Zur Gathen, J.; Viola, A.; Ziegler, K., Counting reducible, powerful, and relatively irreducible multivariate polynomials over finite fields, Siam J. Discrete Math., 27, 855-891, (2013) · Zbl 1348.11094 [17] Wan, D., Zeta functions of algebraic cycles over finite fields, Manuscripta Math., 74, 413-444, (1992) · Zbl 0808.14016
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