Solving difference equations in sequences: Universality and Undecidability

Abstract: $ $We study solutions of difference equations in the rings of sequences and, more generally, solutions of equations with a monoid action in the ring of sequences indexed by the monoid. This framework includes, for example, difference equations on grids (e.g., standard difference schemes) and difference equations in functions on words. On the universality side, we prove a version of strong Nullstellensatz for such difference equations under the assuption that the cardinality of the ground field is greater than the cardinality of the monoid and construct an example showing that this assumption cannot be omitted. On the undecidability side, we show that the following problems are undecidable: $\bullet$ testing radical difference ideal membership or, equivalently, determining whether a given difference polynomial vanishes on the solution set of a given system of difference polynomials; $\bullet$ determining consistency of a system of difference equations in the ring of real-valued sequences; $\bullet$ determining consistency of a system of equations with action of $\mathbb{Z}^2$, $\mathbb{N}^2$, or the free monoid with two generators in the corresponding ring of sequences over any field of characteristic zero.