Big polynomial rings and Stillman’s conjecture

By Daniel Erman, Steven V. Sam, Andrew Snowden. Inventiones Mathematicae, 2018

64 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/be8d2d53b52b8b743a1e3e39fc9510acdc601427

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In Algebra and Number Theory

1. Jan Draisma
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Citations in the corpus

• Topological Noetherianity of polynomial functors (2017, result): “However, in a follow-up paper [17], the same authors give two new proofs of Stillman’s conjecture, one of which uses Corollary 6.”