Small subalgebras of polynomial rings and Stillman’s Conjecture

Abstract:

Let $n, d, \eta$ be positive integers. 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 ${G}_1, \ldots , {G}_{B}$ 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 ${G}_1, \ldots , {G}_{B}$, the ring $R/J$ satisfies the Serre condition $\mathrm {R}_\eta$. 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 $N$.