Equivariant ideals of polynomials
<|reference_start|>Equivariant ideals of polynomials: 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.<|reference_end|>
@article{ghosh2024equivariant,
    title={Equivariant ideals of polynomials},
    author={Arka Ghosh, S{l}awomir Lasota},
    journal={arXiv preprint arXiv:2402.17604},
    year={2024},
    archivePrefix={arXiv},
    eprint={2402.17604},
    primaryClass={cs.LO cs.FL}
}