Computing Galois Groups for Certain Classes of Ordinary Differential Equations

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dc.contributor.advisor Michael Singer, Chair en_US
dc.contributor.advisor Ronald Fulp, Member en_US
dc.contributor.advisor Kailash Misra, Member en_US
dc.contributor.advisor Larry Norris, Member en_US Berman, Peter Hillel en_US 2010-04-02T18:43:14Z 2010-04-02T18:43:14Z 2001-07-25 en_US
dc.identifier.other etd-20010723-022851 en_US
dc.description.abstract As of now, it is an open problem to find an algorithmthat computes the Galois group G of an arbitrary linear ordinary differential operator L in C(x)[D]. We assume thatC is a computable, characteristic-zero,algebraically closed constant field with factorization algorithm.In this dissertation, we present new methods forcomputing differential Galois groups in two special cases.An article by Compoint and Singer presents a decision procedure to compute G in case L is completely reducible or, equivalently, G is reductive. Here, we present the results of an article by Berman and Singerthat reduces the case of a productof two completely reducible operators to thatof a single completely reducible operator;moreover, we give an optimization of that article's core decision procedure.These results rely on results from cohomologydue to Daniel Bertrand.We also give a set of criteria to compute the Galois group of a differential equation of the formy⁽³⁾ + ay' + by = 0, a, b in C[x].Furthermore, we present an algorithm to carry out this computation in case C is the field of algebraic numbers.This algorithm applies the approach used inan article by M. van der Put to study order-two equations with one or two singularpoints. Each step of the algorithm employs a simple, implementable test based on some combination of factorization properties, properties of associated operators,and testing of associated equations for rational solutions. Examples of the algorithm and a Maple implementation writtenby the author are provided. en_US
dc.rights I hereby certify that, if appropriate, I have obtained and attached hereto a written permission statement from the owner(s) of each third party copyrighted matter to be included in my thesis, dissertation, or project report, allowing distribution as specified below. I certify that the version I submitted is the same as that approved by my advisory committee. I hereby grant to NC State University or its agents the non-exclusive license to archive and make accessible, under the conditions specified below, my thesis, dissertation, or project report in whole or in part in all forms of media, now or hereafter known. I retain all other ownership rights to the copyright of the thesis, dissertation or project report. I also retain the right to use in future works (such as articles or books) all or part of this thesis, dissertation, or project report. en_US
dc.title Computing Galois Groups for Certain Classes of Ordinary Differential Equations en_US PhD en_US PhD Dissertation en_US Mathematics en_US

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