We use generating functions to relate the expected values of polynomial factorization statistics over @math to the cohomology of ordered configurations in @math as a representation of the symmetric group. Our methods lead to a new proof of the twisted Grothendieck-Lefschetz formula for squarefree polynomial factorization statistics of Church, Ellenberg, and Farb.
@cite_0: We study the distribution in short intervals of certain arithmetic functions, including the von Mangoldt function and the M "obius function, on polynomials over a finite field @math . Using the Grothendieck-Lefschetz trace formula, we reinterpret each moment of these distributions as a point-counting problem on a highly singular complete intersection variety. We compute part of the @math -adic cohomology of these varieties, yielding an asymptotic bound on each moment for fixed degree @math in the limit as @math .
There have been other generalizations of Theorem from squarefree polynomials to all polynomials. Gadish [Sec. 1.3] Gadish and Hast, Matei @cite_0 both study expected values of functions defined on the set of all polynomials; their functions depend on both the degree of the irreducible factors and their multiplicities. We call these . Gadish [Cor. 1.4] Gadish shows that the expected value of a weighted factorization statistic @math on @math matches the expected value of @math on @math viewed as a class function. Stated geometrically, the expected values of weighted factorization statistics on degree @math polynomials correspond to the cohomology of @math as an @math -representation, while the expected values of our factorization statistics correspond to the cohomology of @math as an @math -representation.