Christian Dönch successfully defended his thesis - Congratulations!
This thesis treats different kinds of standard bases in finitely generated modules over the ring of difference-skew-differential operators, their computation and their application to the computation of multivariate dimension (quasi-)polynomials. It consists of two parts. The first deals with standard bases in modules over the ring of difference-skew-differential operators. The second part deals with uni- and multivariate dimension quasipolynomials associated with such modules.
We start by recalling the notions of skew-differential, difference, and difference-skew-diffe\-rential operators. Skew-differential operators are a generalization of commutative polynomials, differential operators, and difference operators. For the numeric solution of linear differential equations one often considers the associated difference scheme which can be described in terms of inversive difference operators. Combining the notions of skew-differential operators and difference operators we consider difference-skew-differential operators. We present matrix representations for generalized term orders. Then we introduce the notion of weight relative Gröbner bases in finitely generated modules of difference-skew-differential operators generalizing the notions of Gröbner bases, relative Gröbner bases and Gröbner bases with respect to several orderings. We provide a method for their computation. Furthermore we give a characterization of weight relative Gröbner bases. This naturally gives rise to complexity considerations for the aforementioned method. A partial result for commutative polynomials has been presented at ACA 2011. We go on by generalizing the notion of border bases to finitely generated modules of difference-skew-differential operators. We establish a connection between border and Gröbner bases in this setting. Considering multiplication endomorphisms we also derive some S-polynomial-like criteria for a border prebasis to be a border basis. Algorithms for the computation of border bases of zero-dimensional modules are included in the appendix.
We also introduce the notion of weighted filtrations of modules over rings of difference-skew-differential operators and generalize the classical theory of dimension polynomials associated with excellent filtrations to excellent weighted filtrations. We prove the existence of dimension quasipolynomials associated with such excellent weighted filtrations. Considering the module of differentials we can extend this result to differential field extensions. Another extension of our results regards weighted multifiltrations and multivariate dimension functions. Finally we provide several examples for dimension (quasi)polynomials of well-known systems of differential and difference equations from mathematical physics.