RISC PhD Fellowships 2016 [PhD 2016]

@techreport{RISC5387,
author = {N. Thieu Vo},
title = {{Rational and Algebraic Solutions of First-Order Algebraic ODEs}},
language = {english},
abstract = {The main aim of this thesis is to study new algorithms for determining polynomial, rational and algebraic solutions of first-order algebraic ordinary differential equations (AODEs). The problem of determining closed form solutions of first-order AODEs has a long history, and it still plays a role in many branches of mathematics. There is a bunch of solution methods for specific classes of such ODEs. However still no decision algorithm for general first-order AODEs exists, even for seeking specific kinds of solutions such as polynomial, rational or algebraic functions. Our interests are algebraic general solutions, rational general solutions, particular rational solutions and polynomial solutions. Several algorithms for determining these kinds of solutions for first-order AODEs are presented.We approach first-order AODEs from several aspects. By considering the derivative as a new indeterminate, a first-order AODE can be viewed as a hypersurface over the ground field. Therefore tools from algebraic geometry are applicable. In particular, we use birational transformations of algebraic hypersurfaces to transform the differential equation to another one for which we hope that it is easier to solve. This geometric approach leads us to a procedure for determining an algebraic general solution of a parametrizable first-order AODE. A general solution contains an arbitrary constant. For the problem of determining a rational general solution in which the constant appears rationally, we propose a decision algorithm for the general class of first-order AODEs.The geometric method is not applicable for studying particular rational solutions. Instead, we study this kind of solutions from combinatorial and algebraic aspects. In the combinatorial consideration, poles of the coefficients of the differential equation play an important role in the estimation of candidates for poles of a rational solution and their multiplicities. An algebraic method based on algebraic function field theory is proposed to globally estimate the degree of a rational solution. A combination of these methods leads us to an algorithm for determining all rational solutions for a generic class of first-order AODEs, which covers every first-order AODEs from Kamke's collection. For polynomial solutions, the algorithm works for the general class of first-order AODEs.},
number = {16-11},
year = {2016},
month = {12},
note = {Thesis Dissertation},
length = {93},
type = {RISC Report Series},
institution = {Research Institute for Symbolic Computation (RISC), Johannes Kepler University Linz},
address = {Altenberger Straße 69, 4040 Linz, Austria},
issn = {2791-4267 (online)}
}