LAUSR

Ananyan-Hochster's recent proof of Stillman's conjecture reveals a key principle: if $f_1, \dots, f_r$ are elements of a polynomial ring such that no linear combination has small strength then $f_1, \dots, f_r$ behave approximately like independent variables. We show that this approximation becomes in exact in two limits of polynomial rings (the inverse limit and the ultrapower), thereby proving the surprising fact that these limiting rings are themselves polynomial rings (in uncountably many variables). We then use these polynomiality statements to give two new proofs of Stillman's conjecture. The first, via the ultraproduct ring, is similar to the proof of Ananyan--Hochster, but more streamlined. The second, via the inverse limit ring, is totally different, and more geometric in nature.