The geometry of probability generating functions

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The first part of this work concerns the estimation of Taylor coefficients of generating functions with a pole or other singularity along a complex algebraic variety. The relation between the zeros of the polynomial Q and the coefficients of P/Q have been demonstrated in a series of works from 2000 to 2011. Via Cauchy's integral representation, estimating coefficients is reduced to residue integration along the zero set of Q. There are two main open problems addressed in this proposal. In the case where the zero set of Q is smooth, we consider the effective computation of the homology class of the chain of integration with respect to a basis of saddle point contours. In the case where the zero set of Q is singular, only quadratic singularities have as yet been worked out; we consider here two examples of higher degree. The second part of the proposal concerns the relation between the zeros of Q and the coefficients of Q itself. This work is part of the emerging field of stable function theory. The main goal here is to extend the Borcea-Branden-Liggett theory of strong Rayleigh distributions to random variables taking more than two values. A third part of this proposal concerns problems in discrete probability theory, including locating the critical points of a random polynomial or power series, and a problem on totally anisotropic percolation.The broad impact of this research lies in computational infrastructure for computation in combinatorial applications. Generating functions are used in many areas of pure and applied mathematics: random graph theory, theory of algorithms, statistical physics, discrete probability theory, population biology and genetics to name a few. Estimating the coefficients of a generating function is a crucial step in any application of generating function techniques. This work, and particularly the part dealing with automated computation, allows users to compute without having to develop ad hoc methods in each case. Broad impacts of the second and third parts of the proposal include further understanding of negative dependence among random variables. There are also educational outcomes from work on curriculum development and K-12 education enabled by this grant.