Equivariant ideals of polynomials

TL;DR

This work extends classical polynomial ideal theory to rings with infinitely many variables indexed by a countable relational structure ${\mathcal A}$, where embeddings act by renaming variables and equivariant ideals are closed under this action. The authors prove a generalised Hilbert-style Noetherianity: for ω-well-structured, totally ordered ${\mathcal A}$ and any Noetherian ${\mathbb K}$, ${\mathbb K}[{\mathcal A}]$ is equivariantly Noetherian, yielding finite orbit-finite bases for equivariant ideals. They develop a Buchberger-type algorithm to compute Gröbner bases of equivariant ideals from finite presentations of S-polynomials, enabling decidability of ideal membership and broad algorithmic applications. The framework is then applied to orbit-finitely generated vector spaces, weighted register automata zeroness, reversible Petri nets with data, and orbit-finite linear systems, establishing decidability and structural finiteness results in these data-rich, infinite-variable settings. Overall, the paper provides foundational theory and practical algorithms for symmetry-aware polynomial algebra beyond finite-variable regimes, with implications for automata, nets, and linear equation systems in orbit-finite contexts.

Abstract

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 Gröbner 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.

Figures (1)

• Figure 1: $C_5\in\text{\sc Age}(\mathcal{Q}\times\mathcal{Q})$. Its elements are depicted by edges.

Theorems & Definitions (49)

• Example 2
• Example 3
• Example 4
• lemma 1
• Definition 6
• Example 7
• theorem 8
• lemma 2
• Remark 9
• lemma 3