The geometry of probability generating functions

本工作的第一部分是关于沿复代数变化的具有极点或其他奇点的生成函数的泰勒系数的估计。多项式Q的零点与P/Q系数之间的关系已在2000年至2011年的一系列作品中得到证明。通过柯西积分表示，估计系数被简化为沿q的零集的残数积分。在Q的零集是光滑的情况下，考虑了关于鞍点轮廓基的积分链的同调类的有效计算。在Q的零集是奇异的情况下，只有二次奇点被计算出来；我们在这里考虑两个更高程度的例子。该建议的第二部分涉及Q的零点与Q本身的系数之间的关系。这项工作是稳定函数理论这一新兴领域的一部分。这里的主要目标是将强瑞利分布的Borcea-Branden-Liggett理论扩展到取两个以上值的随机变量。本文的第三部分涉及离散概率论中的问题，包括随机多项式或幂级数的临界点定位问题，以及全各向异性渗流问题。本研究的广泛影响在于组合应用中计算的计算基础设施。生成函数用于纯数学和应用数学的许多领域：随机图论、算法理论、统计物理、离散概率论、种群生物学和遗传学等等。在生成函数技术的任何应用中，估计生成函数的系数是至关重要的一步。这项工作，特别是处理自动计算的部分，允许用户无需在每种情况下开发特别的方法即可进行计算。该建议的第二部分和第三部分的广泛影响包括进一步理解随机变量之间的负相关性。在课程开发和K-12教育方面的工作也取得了教育成果。

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.