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：我们研究无限多个变量上的多项式理想有限基的存在性与可计算性。在我们的设定中，变量来源于可数逻辑结构A，且从A到A的嵌入通过重命名变量作用于多项式。首先，我们给出A保证以下希尔伯特基定理推广的充要条件：每个等变的（即在变量重命名下不变）多项式理想都是有限生成的。其次，我们开发了经典Buchberger算法的扩展，用于计算给定等变理想的Gröbner基。这意味着等变理想的成员判定问题是可判定的。最后，我们概述了这些结果在寄存器自动机、带数据的Petri网、轨道有限生成的向量空间以及轨道有限线性方程组中的各种应用。