Semigroup Graded Stillman's Conjecture

TL;DR

This paper proves Stillman's conjecture for polynomial rings graded by semigroups under mild conditions, establishing both sufficiency and necessity of these conditions, with implications for standard graded cases.

Contribution

It extends Stillman's conjecture to semigroup-graded polynomial rings and identifies necessary conditions for the existence of bounds.

Findings

Proves Stillman's conjecture for semigroup-graded polynomial rings.

Identifies necessary conditions for the existence of Stillman bounds.

Provides applications to standard graded polynomial rings.

Abstract

We resolve Stillman's conjecture for families of polynomial rings that are graded by any semigroup under mild conditions. Conversely, we show that these conditions are necessary for the existence of a Stillman bound. This has applications even for the well-known standard graded case.