Quantitative study of semi -Pfaffian sets
Abstract
In the present thesis, we establish upper-bounds on the topological complexity of sets defined using Pfaffian functions. The measures of topological complexity chosen are the various Betti numbers of those sets, and the estimates are given in terms of the natural Pfaffian complexity (also called format) of those sets. Pfaffian functions were introduced by Khovanskii; they are a wide class of real-analytic functions which are solutions of certain polynomial differential systems and have polynomial-like behaviour over the real domain. Semi-Pfaffian sets are sets that satisfy a quantifier-free sign condition on such functions, and sub-Pfaffian sets are linear projection of semi-Pfaffian sets. Pfaffian functions generate an o-minimal structure, and Gabrielov showed that this structure could be described by Pfaffian limit sets, which are constructed using limits of 1-parameter semi-Pfaffian families. We give effective estimates for all of the above cases. The techniques used are standard Morse theory, deformations of low complexity, recursion on combinatorial levels and a spectral sequence for continuous surjections.
Degree
Ph.D.
Advisors
Gabrielov, Purdue University.
Subject Area
Mathematics
Off-Campus Purdue Users:
To access this dissertation, please log in to our
proxy server.