Using recent work by Erman-Sam-Snowden, we show that finitely generated
ideals in the ring of bounded-degree formal power series in infinitely many
variables have finitely generated Gr\"obner bases relative to the graded
reverse lexicographic order. We then combine this result with the first
author's work on topological Noetherianity of polynomial functors to give an
algorithmic proof of the following statement: ideals in polynomial rings
generated by a fixed number of homogeneous polynomials of fixed degrees only
have a finite number of possible generic initial ideals, independently of the
number of variables that they involve and independently of the characteristic
of the ground field. Our algorithm outputs not only a finite list of possible
generic initial ideals, but also finite descriptions of the corresponding
strata in the space of coefficients.Comment: Several minor edit