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 (for example, 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 assumption 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:

中文翻译：

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求解序列中的微分方程：普遍性和不可判定性

我们研究序列环中差分方程的解，更一般地，研究在由幺半群索引的序列环中具有幺半群作用的方程的解。例如，该框架包括网格上的差分方程（例如，标准差分方案）和单词函数中的差分方程。在普遍性方面，我们在地面场的基数大于幺半群的基数的假设下证明了此类差分方程的强 Nullstellensatz 版本，并构造了一个示例，表明该假设不能省略。在不可判定性方面，我们表明以下问题是不可判定的：