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 I I of R R whose generators are forms of degree ≤ d \leq d is bounded independent of N N . We also show that there is a primary decomposition of I 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 N .

Let n, d, η 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 ≤ η B(n, d) forms G 1 , . . . , G B of degree ≤ d, where η 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 Rη. These results imply a conjecture of M. Stillman asserting that the projective dimension of an n-generator ideal of R whose generators are forms of degree ≤ d is bounded independent of N . We also show that there is a primary decomposition of the ideal 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 .

Let R be a polynomial ring in N variables over an arbitrary field K and let I be an ideal of R generated by n polynomials of degree at most 2. We show that there is a bound on the projective dimension of R/I that depends only on n, and not on N . The proof depends on showing that if K is infinite and n is a positive integer, there exists a positive integer C(n), independent of N , such that any n forms of degree at most 2 in R are contained in a subring of R generated over K by at most t ≤ C(n) forms G1, . . . , Gt of degree 1 or 2 such that G1, . . . , Gt is a regular sequence in R. C(n) is asymptotic to 2n2n .

In [2], the authors prove Stillman's conjecture in all characteristics and all degrees by showing that, independent of the algebraically closed field K or the number of variables, n forms of degree at most d in a polynomial ring R over K are contained in a polynomial subalgebra of R generated by a regular sequence consisting of at most η B(n, d) forms of degree at most d: we refer to these informally as "small" subalgebras. Moreover, these forms can be chosen so that the ideal generated by any subset defines a ring satisfying the Serre condition Rη. A critical element in the proof is to show that there are functions η A(n, d) with the following property: in a graded n-dimensional K-vector subspace V of R spanned by forms of degree at most d, if no nonzero form in V is in an ideal generated by η A(n, d) forms of strictly lower degree (we call this a strength condition), then any homogeneous basis for V is an Rη sequence. The methods of [2] are not constructive. In this paper, we use related but different ideas that emphasize the notion of a key function to obtain the functions η A(n, d) in degrees 2, 3, and 4 (in degree 4 we must restrict to characteristic not 2, 3). We give bounds in closed form for the key functions and the η A functions, and explicit recursions that determine the functions η B from the η A functions. In degree 2, we obtain an explicit value for η B(n, 2) that gives the best known bound in Stillman's conjecture for quadrics when there is no restriction on n. In particular, for an ideal I generated by n quadrics, the projective dimension R/I is at most 2 n+1 (n − 2) + 4.Theorem 1.1. There is an upper bound, independent of N , on pd R (R/I), where I is any ideal of R generated by n homogeneous polynomials F 1 , . . . , F n of given degrees d 1 , . . . , d n .We refer to such bounds, which are now known to exist, as Stillman bounds.

In an earlier work, the authors prove Stillman’s conjecture in all characteristics and all degrees by showing that, independent of the algebraically closed field K K or the number of variables, n n forms of degree at most d d in a polynomial ring R R over K K are contained in a polynomial subalgebra of R R generated by a regular sequence consisting of at most η B ( n , d ) {}^\eta \!B(n,d) forms of degree at most d d ; we refer to these informally as “small” subalgebras. Moreover, these forms can be chosen so that the ideal generated by any subset defines a ring satisfying the Serre condition R η _\eta . A critical element in the proof is to show that there are functions η A ( n , d ) {}^\eta \!A(n,d) with the following property: in a graded n n -dimensional K K -vector subspace V V of R R spanned by forms of degree at most d d , if no nonzero form in V V is in an ideal generated by η A ( n , d ) {}^\eta \!A(n,d) forms of strictly lower degree (we call this a strength condition), then any homogeneous basis for V V is an R η _\eta sequence. The methods of our earlier work are not constructive. In this paper, we use related but different ideas that emphasize the notion of a key function to obtain the functions η A ( n , d ) {}^\eta \!A(n,d) in degrees 2, 3, and 4 (in degree 4 we must restrict to characteristic not 2, 3). We give bounds in closed form for the key functions and the η A _ {}^\eta \!{\underline {A}} functions, and explicit recursions that determine the functions η B {}^\eta \!B from the η A _ {}^\eta \!{\underline {A}} functions. In degree 2, we obtain an explicit value for η B ( n , 2 ) {}^\eta \!B(n,2) that gives the best known bound in Stillman’s conjecture for quadrics when there is no restriction on n n . In particular, for an ideal I I generated by n n quadrics, the projective dimension R / I R/I is at most 2 n + 1 ( n − 2 ) + 4 2^{n+1}(n - 2) + 4 .