Dissertations, Theses, and Capstone Projects

Date of Degree

10-2014

Document Type

Dissertation

Degree Name

Ph.D.

Program

Mathematics

Advisor

Alexey Ovchinnikov

Subject Categories

Mathematics

Keywords

Differential Galois theory, Linear differential algebraic groups, Parameterized Picard-Vessiot theory

Abstract

We present algorithms to compute the differential Galois group G associated via the parameterized Picard-Vessiot theory to a parameterized second-order linear differential equation with respect to d/dx, with coefficients in the field of rational functions F(x) over a differential field F, where we think of the derivations on F as being derivations with respect to parameters. We build on an earlier procedure, developed by Dreyfus, that computes G when the equation is unimodular, assuming either that G is reductive, or else that its maximal reductive quotient is differentially constant. We first show how to modify the space of parametric derivations to reduce the computation of the unipotent radical of G to the case when the reductive quotient is differentially constant in the unimodular case. For non-unimodular equations, we reinterpret a classical change-of-variables procedure in Galois-theoretic terms in order to reduce the computation of G to the computation of an associated unimodular differential Galois group H. We establish a parameterized version of the Kolchin-Ostrowski theorem and apply it to give more direct proofs than those found in the literature of the fact that the required computations can be performed effectively. We then extract from these algorithms a complete set of criteria to decide whether any of the solutions to a parameterized second-order linear differential equation is differentially transcendental with respect to the parametric derivations. We give various examples of computation and some applications to differential transcendence

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