# Equivariant ideals of polynomials

arXiv: https://arxiv.org/abs/2402.17604 · HTML: https://mlsanity.com/paper/2402.17604/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Feb 2024.

## Built on by (most cited first, 2 of 2)

- [Bi-reachability in Petri nets with data](https://mlsanity.com/paper/2405.16176/) (2024)
- [Reachability in Symmetric VASS](https://mlsanity.com/paper/2506.23578/) (2025)

## Abstract (arXiv)

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.