Gröbner Bases in Partial Differential Equations

Summary

1. In this paper, I will describe some recent (and rather unexpected) applications of the theory of Gröbner bases to the study of the structure of solutions of linear systems of constant coefficients partial differential systems. Gröbner bases first appeared in Buchbergers's Ph.D. thesis [5] (see also [6] where the main results were first published), and their theory has provided the conceptual basis for the creation of several computational algebra packages which can be utilized for the solution of polynomial problems. The approach that I have successfully applied in a series of joint papers [1], [2], [3], [4], [10], [11] is made possible by the algebrization of analysis which began in the sixties [12], [16] and was then perfected by M. Sato and his collaborators in the seventies [20], [21]. In this introductory section, I will briefly recall the foundations of this algebraic approach to partial differential equations, while in section 2, I will show a few concrete and remarkable applications of Gröbner bases to specific systems of differential equations. I would like to point out that the way in which we have been using Gröbner bases is twofold: on one hand we have used some symbolic computation packages which are based on the theory of Gröbner bases; on the other hand (see Theorem 2), we have used the theory itself to generalize results which had been computed in special cases.