Algorithms for bivariate singularity analysis / Timothy DeVries.

Format:

Book

Manuscript

Thesis/Dissertation

Author/Creator:

DeVries, Timothy (Timothy John)

Contributor:

Pemantle, Robin, advisor.

University of Pennsylvania.

Language:

English

Subjects (All):

Penn dissertations--Mathematics.

Mathematics--Penn dissertations.

Local Subjects:

Penn dissertations--Mathematics.

Mathematics--Penn dissertations.

Physical Description:

vii, 125 pages ; 29 cm

Production:

2011.

Summary:

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.

Notes:

Adviser: Robin Pemantle.

Thesis (Ph.D. in Mathematics) -- University of Pennsylvania, 2011.

Includes bibliographical references.