| Abstract: |
| We present a homogeneous Groebner theoretic framework for E*-Groebner bases in difference modules and applications to bivariate difference dimension polynomials. The approach uses graded homogenization in which the homogenizing variable z is taken to be the greatest variable. With this choice homogeneous leading terms encode affine coleaders and homogeneous reduction dehomogenizes naturally to E*-reduction. This yields a Buchberger type algorithm for E*-Groebner bases via homogenization and saturation placing effective order methods into a standard Groebner basis framework.
Using these bases we construct bivariate difference dimension polynomials for finitely generated difference modules and difference field extensions. While the classical univariate dimension polynomial measures total order growth the bivariate polynomial describes how growth is distributed across order windows. We prove that certain coefficients are generator independent invariants refining the classical theory. These invariants distinguish nonisomorphic difference modules and detect nonequivalence of systems sharing identical invariants of univariate dimension polynomials. |
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