Date of Award
Spring 2011
Degree Type
Dissertation
Degree Name
Doctor of Philosophy (PhD)
Graduate Group
Mathematics
First Advisor
Robin Pemantle
Abstract
An algorithm for bivariate singularity analysis is developed. For a wide class of bivariate, rational functions F = P/Q, this algorithm produces rigorous numerics for the asymptotic analysis of the Taylor coefficients of F at the origin. The paper begins with a self-contained treatment of multivariate singularity analysis. The analysis itself relies heavily on the geometry of the pole set VQ of F with respect to a height function h. This analysis is then applied to obtain asymptotics for the number of bicolored supertrees, computed in a purely multivariate way. This example is interesting in that the asymptotics can not be computed directly from the standard formulas of multivariate singularity analysis. Motivated by the topological study required by this example, we present characterization theorems in the bivariate case that classify the geometric features salient to the analysis. These characterization theorems are then used to produce an algorithm for this analysis in the bivariate case. A full implementation of the algorithm follows.
Recommended Citation
DeVries, Timothy, "Algorithms for Bivariate Singularity Analysis" (2011). Publicly Accessible Penn Dissertations. 326.
https://repository.upenn.edu/edissertations/326