Generalizations of Stillman's conjecture via twisted commutative algebras

TL;DR

This paper generalizes Stillman's conjecture by leveraging recent advances in twisted commutative algebras, establishing broad boundedness results in commutative algebra that depend solely on generator degrees.

Contribution

It extends Stillman's conjecture to a wider context using twisted commutative algebra, providing new boundedness results independent of the number of variables.

Findings

Proves a broad generalization of Stillman's conjecture.

Establishes boundedness results based only on generator degrees.

Utilizes recent noetherianity results in twisted commutative algebras.

Abstract

Combining recent results on noetherianity of twisted commutative algebras by Draisma and the resolution of Stillman's conjecture by Ananyan-Hochster, we prove a broad generalization of Stillman's conjecture. Our theorem yields an array of boundedness results in commutative algebra that only depend on the degrees of the generators of an ideal, and not the number of variables in the ambient polynomial ring.