1,684 research outputs found
On Shifted Eisenstein Polynomials
We study polynomials with integer coefficients which become Eisenstein
polynomials after the additive shift of a variable. We call such polynomials
shifted Eisenstein polynomials. We determine an upper bound on the maximum
shift that is needed given a shifted Eisenstein polynomial and also provide a
lower bound on the density of shifted Eisenstein polynomials, which is strictly
greater than the density of classical Eisenstein polynomials. We also show that
the number of irreducible degree polynomials that are not shifted
Eisenstein polynomials is infinite. We conclude with some numerical results on
the densities of shifted Eisenstein polynomials
On the p-parts of Weyl group multiple Dirichlet series
We study the structure of -parts of Weyl group multiple Dirichlet series.
In particular, we extend results of Chinta, Friedberg, and Gunnells and show,
in the stable case, that the -parts of Chinta and Gunnells agree with those
constructed using the crystal graph technique of Brubaker, Bump, and Friedberg.
In this vein, we give an explicit recurrence relation on the coefficients of
the -parts, which allows us to describe the support of the -parts and
address the extent to which they are uniquely determined.Comment: 18 pages, 4 figures; typos corrected, extended results in section
Quasimodularity and large genus limits of Siegel-Veech constants
Quasimodular forms were first studied in the context of counting torus
coverings. Here we show that a weighted version of these coverings with
Siegel-Veech weights also provides quasimodular forms. We apply this to prove
conjectures of Eskin and Zorich on the large genus limits of Masur-Veech
volumes and of Siegel-Veech constants.
In Part I we connect the geometric definition of Siegel-Veech constants both
with a combinatorial counting problem and with intersection numbers on Hurwitz
spaces. We introduce modified Siegel-Veech weights whose generating functions
will later be shown to be quasimodular.
Parts II and III are devoted to the study of the quasimodularity of the
generating functions arising from weighted counting of torus coverings. The
starting point is the theorem of Bloch and Okounkov saying that q-brackets of
shifted symmetric functions are quasimodular forms. In Part II we give an
expression for their growth polynomials in terms of Gaussian integrals and use
this to obtain a closed formula for the generating series of cumulants that is
the basis for studying large genus asymptotics. In Part III we show that the
even hook-length moments of partitions are shifted symmetric polynomials and
prove a formula for the q-bracket of the product of such a hook-length moment
with an arbitrary shifted symmetric polynomial. This formula proves
quasimodularity also for the (-2)-nd hook-length moments by extrapolation, and
implies the quasimodularity of the Siegel-Veech weighted counting functions.
Finally, in Part IV these results are used to give explicit generating
functions for the volumes and Siegel-Veech constants in the case of the
principal stratum of abelian differentials. To apply these exact formulas to
the Eskin-Zorich conjectures we provide a general framework for computing the
asymptotics of rapidly divergent power series.Comment: 107 pages, final version, to appear in J. of the AM
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