Strength conditions, small subalgebras, and Stillman bounds in degree $\leq 4$

Abstract

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.

Properties of high rank subvarieties of affine spaces

“…The part ( 4) is an immediate consequence of the part (3). (3) As shown in [1] the part (4) of the theorem implies the validity of an effective Stillman conjecture.…”

“…Conjecture 3.5. The dependence of r on s in (1) is polynomial for char(k) > d, namely we have r = s −O d (1) . The conjecture is known for d = 2, 3, 4 ( [11], [22]).…”

“…To prove(1) we show that the existence of non-zero elements G, H ∈ S such that G H = 0 leads to a contradiction. If G H = 0 then GH = 0 where G, H are the top homogeneous terms of G, H. Factor the highest possible powers of F from G and H. So G = F i G ′ , H = F j H ′ where G ′ and H ′ are not divisible by F .…”

Universal Lex Ideal Approximations of Extended Hilbert Functions and Hamilton Numbers

“…2 ) is lexicographic in R (2) but not in R (3) , because in the larger ring x 1 x 3 > x 2 2 . We shall say that an ideal of R (h) is a universal lex ideal if its extension to R (N ) is lexicographic for all N ≥ h. The usage of universal is the same as in [12].…”

“…There has been a great deal of work recently (see, for example, [1,2,3,5,7,8,9,15,16,22] and their references) on the behavior of invariants of ideals generated in at most a given degree and with at most a given number of generators when the number of variables is not bounded in any way. The problems we study were motivated by a related question described below.…”