Strength conditions, small subalgebras, and Stillman bounds in degree ≤4

Abstract

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 .