We establish several finiteness properties of groups defined by algebraic difference equations. One of our main results is that a subgroup of the general linear group defined by possibly infinitely many algebraic difference equations in the matrix entries can indeed be defined by finitely many such equations. As an application, we show that the difference ideal of all difference algebraic relations among the solutions of a linear differential equation is finitely generated.

中文翻译：

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仿射差分代数群的有限性性质

我们建立了由代数差分方程定义的群的几个有限性质。我们的主要结果之一是，由矩阵项中可能无限多个代数差分方程定义的一般线性群的子群确实可以由有限多个这样的方程定义。作为一个应用，我们证明了线性微分方程解之间所有差分代数关系的差分理想是有限生成的。