Habilitationen in der Computeralgebra

Abstract

This thesis approaches partial differential equations (PDEs) from the viewpoint of algebra and contributes algorithmic methods which allow to investigate effectively the relationship of systems of PDEs and their sets of solutions. Employing formal techniques, the focus is on polynomial differential equations and their analytic solutions. We borrow quite a few concepts from algebraic geometry. Whenever a set of points is given by a polynomial or rational parametrization, an elimination of the parameters from the equations which express the coordinates of the points yields equations that are satisfied by the coordinates of every point of the set. If there exists an implicit description of this set as solution set of a system of polynomial equations, then elimination constructs such a description. This thesis develops algorithmic methods which accomplish the analogous elimination task for systems of polynomial partial differential equations and their (complex) analytic solutions. It builds on work by C. Riquier, M. Janet, J. M. Thomas, J. F. Ritt, E. R. Kolchin, and others, who laid the foundation of differential algebra. A given multivariate polynomial, whose coefficients are analytic functions, is interpreted as a parametrization of a set of analytic functions, i.e., every element of this set arises from substitution of appropriate analytic functions for the indeterminates of the polynomial. Moreover, the substitution of functions for the indeterminates also involves the composition with prescribed analytic functions. If the polynomial is linear, then the resulting set is a vector space over the field of constants. In general, however, the parametrized set is rarely