Summary: APPLICATIONS OF COMMUTATIVE AND COMPUTATIONAL ALGEBRA TO
PARTIAL DIFFERENTIAL EQUATIONS
W.W. ADAMS, P. LOUSTAUNAU, D.C. STRUPPA
Abstract. We propose a computational algebra approach to some classical problems in the theory of linear
partial differential equations. Our approach is based on the developments of new tools in computational
algebra, as well as on the algebraization process which has occured in the theory of partial differential
equations over the last 30 years. We describe several problems in well­known areas (calculation of Ext
modules, free resolutions, Koszul complexes, Noetherian operators, etc.) but we also describe some new
problems such as the development of a possible theory for weakly commutative systems.
In this paper we propose a new approach to some problems in partial differential equations based on compu­
tational algebra. We list some fundamental problems whose solutions would validate our approach. We hope
that this paper will pave the way towards research in computational algebraic analysis. A more detailed
version of this work will be published subsequently. We also refer to [2, 3, 4], where we show how some of
our ideas can be effectively implemented in some concrete cases.
Let us first describe the general set up for our approach. Let R = C [z 1 ; : : : ; z n ], and let P = [P ij ] be
an r 1 \Theta r 0 matrix of polynomials in R so that P (D) = [P ij (D)] is a matrix of linear constant coefficients
differential operators, where D =
i
\Gammai @
@x1 ; : : : ; \Gammai @

Source: Adams, William W. - Department of Mathematics, University of Maryland at College Park

Collections: Mathematics