Structure Analysis of Polynomial Modules

Abstract

We now apply the theory of involutive bases developed in Chapter 3 to the structure analysis of modules over the commutative polynomial ring, i. e. we do some classical commutative algebra (the question to what extent the results presented here also hold for more general polynomial rings of solvable type is still open and will not be discussed here). The basic observation is that the Pommaret basis with respect to the degree lexicographic term order provides us with an easy access to several interesting invariants; more precisely, this basis is to a considerable extent determined by the structure of the module. As this particular type of basis will also play an important role in the analysis of differential equations, this fact simultaneously allows us an algebraic interpretation of many aspects of the theory of differential equations.

We start in the first section by analysing combinatorial or Stanley decompositions: a polynomial module is written as a direct sum of free modules over polynomial rings in a restricted set of variables. It is a natural consequence of our definition of involutive bases (and in fact the main motivation for it) that any involutive basis of a submodule of a free polynomial module immediately induces such a decomposition. For more general modules the situation is more complicated. Assuming that we deal with a finitely generated module, we can present it as the quotient of a free polynomial module by a submodule and then construct a complementary decomposition to a Gröbner basis of the submodule.