Bivariate Dimension Polynomials of Non-Reflexive Prime Difference-Differential Ideals.: The Case of One Translation

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.