Séminaire Lotharingien de Combinatoire, 80B.6 (2018), 12 pp.
Trevor Hyde
Factorization Statistics and the Twisted
Grothendieck-Lefschetz formula
Abstract.
Factorization statistics are functions defined on the set
Polyd(Fq) of all monic
degree d polynomials with coefficients in
Fq which only depend on the degrees of the irreducible factors of a
polynomial. We show that the expected values of factorization statistics
are determined by the representation theoretic structure of the
cohomology of point configurations in R3. This twisted
Grothendieck-Lefschetz formula for Polyd is analogous to a result
of Church, Ellenberg, and Farb for squarefree polynomials. Our
proof uses formal power series methods which also lead to a new proof
of the Church, Ellenberg, and Farb result circumventing algebraic
geometry.
Received: November 14, 2017.
Accepted: February 17, 2018.
Final version: April 1, 2018.
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