We study existence and computability of finite bases for ideals of polynomials over infinitely many variables. In our setting, variables come from a countable logical structure A, and embeddings from A to A act on polynomials by renaming variables. First, we give a sufficient and necessary condition for A to guarantee the following generalisation of Hilbert's Basis Theorem: every polynomial ideal which is equivariant, i.e. invariant under renaming of variables, is finitely generated. Second, we develop an extension of classical Buchberger's algorithm to compute a Gr\"obner basis of a given equivariant ideal. This implies decidability of the membership problem for equivariant ideals. Finally, we sketch upon various applications of these results to register automata, Petri nets with data, orbit-finitely generated vector spaces, and orbit-finite systems of linear equations.

翻译：我们研究无限多变量多项式理想的有限基存在性与可计算性。在此设定中，变量来源于可数逻辑结构A，且从A到A的嵌入通过变量重命名作用于多项式。首先，我们给出A满足以下希尔伯特基定理推广形式的充要条件：每个等变多项式理想（即在变量重命名下不变的理想）都是有限生成的。其次，我们发展了经典布赫伯格算法的扩展，用于计算给定等变理想的Gröbner基。这蕴含了等变理想成员判定问题的可判定性。最后，我们概述了这些结果在寄存器自动机、带数据的佩特里网、轨道有限生成向量空间以及轨道有限线性方程组中的多种应用。