Grobner bases for ideals in Laurent polynomial rings and their applicationto systems of difference equations

We develop a basic theory of Grobner bases for ideals in the algebra of Lau rent polynomials (and, more generally, in its monomial subalgebras). For th is we have to generalize the notion of term order. The theory is applied to systems of linear partial difference equations (with constant coefficients ) on Z(n). Furthermore, we present a method to compute the intersection of an ideal in the algebra of Laurent polynomials with the subalgebra of all p olynomials.