Polynomial ideals and associated differential operators

In a natural way we can associate to ideals in polynomial rings (in n variables, with coefficients in a field of characteristic 0) certain modules (over these polynomial rings) of differential operators. In this project this correspondence shall be studied in detail (using methods of commutative algebra, linear algebra and symbolic computation). An algorithm to compute a system of generators of these modules of differential operators shall be developed and implemented in MAPLE. For certain ideals with (Krull-) dimension 0 such an algorithm has been developed by Marinari, Möller and Mora. A "dictionary" shall be elaborated to enable us to read off properties of an ideal from properties of the corresponding module of differential operators (and vice versa). This problem (and its solution in a special case) goes back to W. Gröbner, who thus wanted to solve problems of commutative algebra by methods of analysis. Later it was taken up by other authors in view of applications in the fields of multivariate interpolation, partial differential equations and multivariate system theory. Noetherian operators of an ideal, studied by Palamodov and Oberst, are in the case of dimension 0 a system of generators of the module of associated differential operators. In the case of positive dimension the relations between Noetherian operators and these modules are still to be clarified.

It is well known that a subspace of a finite-dimensional vector space can be given either by a finite basis or by a system of linear equations. In almost the same manner an ideal of a polynomial ring can be described either through a finite system of generators or by its "inverse system", introduced by F. Macaulay almost hundred years ago. Elements of the inverse system can be viewed as differential operators. More precisely, let I be an ideal in a polynomial ring F[s]:= F[s 1 ,, s n ] in n variables over an arbitrary field F. The inverse system of I is the vector space of all F-linear functions from F[s] to F mapping the ideal I to 0. In this project the inverse system of zero-dimensional ideals (i.e. ideals with the property that the corresponding system of polynomial equations has only finitely many solutions) was studied in detail. Using the theory of Gröbner bases a method to obtain a basis of the inverse system seen as an F-vector space (in this way the inverse system is described by a finite set of data) was developed. This method does not require the computation of the zeroes of I. Moreover an algorithm to compute a minimal system of generators of the inverse system (seen as an F[s]-module) was developed (such a system of generators admits a more compact description of the inverse system). As an application of the inverse system, the well known (and frequently used) notion of squarefree decomposition of univariate polynomials was generalised to zero-dimensional polynomial ideals and a method to compute this decomposition was conceived. As in the case for univariate polynomials this method does not require the (intricate and frequently practically impossible) computation of the zeroes of the ideal. All algorithms developed have been implemented in the computer-algebra system Maple 8. A software-library with detailed annotations was compiled.