Solving difference equations in sequences: Universality and Undecidability

TL;DR

This paper explores the solutions of difference equations in sequence rings, establishing a strong Nullstellensatz under certain conditions and proving the undecidability of key problems like ideal membership and system consistency.

Contribution

It proves a version of the Nullstellensatz for difference equations in sequence rings and demonstrates the undecidability of fundamental problems in this context.

Findings

Strong Nullstellensatz for difference equations with large ground fields

Undecidability of radical ideal membership testing

Undecidability of system consistency in sequence rings

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.