Algorithms in computational algebraic analysis

This thesis studies algorithms for symbolic computation of systems of linear partial differential equations using the corresponding ring of linear differential operators with polynomial coefficients, which is called the Weyl algebra An. Bernstein-Sato polynomials, one of the central notions in the algebraic analysis of D-modules, is the topic of the first part of this work. We consider the question of constructibility of the stratum of polynomials of bounded number of variables and degree that produce a fixed Bernstein-Sato polynomial. Not only do we give a positive answer, but we construct an algorithm for computing these strata. Another theme of this thesis is two theorems of Stafford that say that every (left) ideal of An can be generated by two elements, and every holonomic An-module is cyclic, i.e. generated by one element. We reprove these results in an effective way that leads to algorithms for computation of these generators. The main engine of all our algorithms is Grobner bases computations in the Weyl algebra. In order to speed these up we developed a parallel version of a Buchberger algorithm, which has been implemented and tested out using supercomputers and has delivered impressive speedups on several important examples.