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.
The first moment formula is derived from the Grothendieck-Lefschetz trace formula for ' e tale cohomology with twisted coefficients'' from which the name is borrowed. The author and Lagarias @cite_2 use Theorem to establish a representation theoretic interpretation of the @math , where @math is the probability of a random squarefree polynomial having factorization type @math .
@cite_2
2786358643
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.