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I am interested in studying ideals of an infinite polynomial ring $k[\dots,x_{-2},x_{-1},x_0,x_1\,\dots]$ which are invariant under translation of variables, e.g. if $x_3^2x_1^3-x_2^4$ belongs to such an ideal, then also $x_{3+n}^2x_{1+n}^3-x_{2+n}^4$ belongs to it for all $n\in\mathbb Z$.

Symmetric ideals, namely ideals invariant under any permutation of variables, have been studied for example in Nagpal and Snowden - Symmetric ideals of the infinite polynomial ring and Aschenbrenner and Hillar - An Algorithm for Finding Symmetric Gröbner Bases in Infinite Dimensional Rings. However I can't find references for ideals invariant under more general actions on the variables (and in particular translations). Where can I find them, if there are?

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This reminds me of Inc-invariant ideals, see Juhnke-Kubitzke, Dinh Van Le, and Römer - Asymptotic behavior of symmetric ideals: A brief survey and references therein.

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You might be interested in "Gröbner bases of ideals invariant under endomorphisms", by Vesselin Drensky and Roberto La Scala, Journal of Symbolic Computation (2006), Issue 7, Pages 835-846. In your specific situation, the general theory might be able to say something particularly useful.

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