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Equivariant polynomial ideals and their membership problem
Abstract:
For an infinite set X (of variables) and a group G acting on X, an ideal J in the polynomial ring K[X] in X over the field K is called equivariant (w.r.t. G) if J is invariant under the action of G. We give a necessary and sufficient condition on the action of G on X for the following extension of Hilbert's basis theorem: every equivariant ideal in K[X] is finitely generated. We also prove that this condition implies decidability of the equivariant ideal membership problem: given a polynomial f and finite set of polynomials H, decide whether f is in the equivariant ideal generated by H.
These results in particular imply decidability of the zeroness problem of weighted register automata, of the universality problem of unambiguous register automata without guessing, and of the reachability problem of reversible data Petri nets for a wide range of data domains.
This is a joint work with Sławomir Lasota and Aliaume Lopez.

English
https://u-paris.zoom.us/j/82303001469?pwd=xapukD4PTaOrYkXkNJNeVLyC4FH7mX.1