Equivariant ideals of polynomials
CoRR(2024)
摘要
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.
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