We use the method of characteristic sets with respect to two term orderings to prove the existence and obtain a method of computation of a bivariate dimension polynomial associated with a non-reflexive difference-differential ideal in the algebra of difference-differential polynomials with several basic derivations and one translation. As a consequence, we obtain a new proof and a method of computation of the dimension polynomial of a non-reflexive prime difference ideal in the algebra of difference polynomials over an ordinary difference field. We also discuss applications of our results to systems of algebraic difference-differential equations.

We introduce a method of characteristic sets with respect to several term orderings for difference-differential polynomials. Using this technique, we obtain a method of computation of multivariate dimension polynomials of finitely generated difference-differential field extensions. Furthermore, we find new invariants of such extensions and show how the computation of multivariate difference-differential polynomials is applied to the equivalence problem for systems of algebraic difference-differential equations.

Let D be a ring of Ore polynomials in m variables x 1 ,...,x m over a field K and let a partition of the set {x 1 ,...,x m}into p disjoint subsets be fixed, so that D can be treated as a filtered ring with the natural p-dimensional ltration associated with the partition.We introduce a special type of reduction in a finitely generated free D-module and develop the corresponding Gröbner basis technique that allows one to prove the existence and find invariants of a dimension polynomial in p variables associated with a finitely generated D-module. We also outline a method of computation of such a polynomial and obtain an essential generalization of the Kolchin theorem on the dimension polynomial of a differential field extension.

MOTIVATION: With the main focus of the Human Genome Project shifting to sequencing, bioinformatics support for constructing large-scale genomic maps of other organisms is still required. We attempt to provide for this with our work, aimed at the delivery of robust and user-friendly contig-building software on the WWW. RESULTS: We present a prototype distributed analytical environment for molecular biologists working in the area of genomic mapping. It consists of the WWW server for constructing contigs from users' data with a hypertext output connected to Java-based map visualization software. AVAILABILITY: Freely available on http://www.mpimg-berlin-dahlem.mpg. de/ approximately andy/server/ CONTACT: [email protected]