A characteristic set method for ordinary difference polynomial systems

We prove several basic properties for difference ascending chains, including a necessary and sufficient condition for an ascending chain to be the characteristic set of its saturation ideal and a necessary and sufficient condition for an ascending chain to be the characteristic set of a reflexive prime ideal. Based on these properties, we propose an algorithm to decompose the zero set of a finite set of difference polynomials into the union of zero sets of certain ascending chains. This decomposition algorithm is implemented and used to solve the perfect ideal membership problem, and to prove certain difference identities automatically.