Small Subalgebras of Polynomial Rings and Stillman's Conjecture

TL;DR

This paper proves bounds on the structure and properties of subalgebras generated by forms of bounded degree in polynomial rings, confirming Stillman's conjecture that projective dimension is independent of the number of variables.

Contribution

It establishes uniform bounds on subalgebra generators and primary decompositions, confirming Stillman's conjecture across arbitrary characteristic fields.

Findings

Bound on the number of generators in subalgebras independent of variables

Existence of regular sequences with bounded degree forms

Primary decomposition invariants are uniformly bounded

Abstract

We show that in a polynomial ring $R$ in $N$ variables over an algebraically closed field $K$ of arbitrary characteristic, any $K$-subalgebra of $R$ generated over $K$ by at most $n$ forms of degree at most $d$ is contained in a $K$-subalgebra of $R$ generated by $B \leq {}^\eta\mathcal{B}(n,d)$ forms $G_1,..., G_B$ of degree $\leq d$, where ${}^\eta\mathcal{B}(n,d)$ does not depend on $N$ or $K$, such that these forms are a regular sequence and such that for any ideal $J$ generated by forms that are in the $K$-span of $G_1, ..., G_B$, the ring $R/J$ satisfies the Serre condition $R_\eta$. These results imply a conjecture of M. Stillman asserting that the projective dimension of an $n$-generator ideal $I$ of $R$ whose generators are forms of degree $\leq d$ is bounded independent of $N$. We also show that there is a primary decomposition of $I$ such that all numerical invariants of the decomposition (e.g., the number of primary components and the degrees and numbers of generators of all of the prime and primary ideals occurring) are bounded independent of $N$.