Small subalgebras of polynomial rings and Stillman’s Conjecture

Let n , d , η n, d, \eta be positive integers. We show that in a polynomial ring R R in N N variables over an algebraically closed field K K of arbitrary characteristic, any K K -subalgebra of R R generated over K K by at most n n forms of degree at most d d is contained in a K K -subalgebra of R R generated by B ≤ η B ( n , d ) B \leq {}^\eta \mathcal {B}(n,d) forms G 1 , … , G B {G}_1,\,\ldots ,\,{G}_{B} of degree ≤ d \leq d , where η B ( n , d ) {}^\eta \mathcal {B}(n,d) does not depend on N N or K K , such that these forms are a regular sequence and such that for any ideal J J generated by forms that are in the K K -span of G 1 , … , G B {G}_1,\,\ldots ,\,{G}_{B} , the ring R / J R/J satisfies the Serre condition R η \mathrm {R}_\eta . These results imply a conjecture of M. Stillman asserting that the projective dimension of an n n -generator ideal