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 .

Universality of High-Strength Tensors

“…=: e has the property that P (e)p = q (1) + q (2) + • • • = q, as desired. ∞ are linearly independent and p ∈ P ∞ has a dense GL ∞ -orbit.…”

“…The strength of polynomials plays a key role in the resolution of Stillman's conjecture by Ananyan-Hochster [1,2], the subsequent work by Erman-Sam-Snowden [12][13][14] and in Kazhdan-Ziegler's work [19,20]. Also see [3-5, 7, 9, 10] for other recent papers studying strength.…”

Universality of high-strength tensors

“…The strength of polynomials plays a key role in the resolution of Stillman's conjecture by Ananyan-Hochster [1,2], the subsequent work by Erman-Sam-Snowden [12,13,14] and in Kazhdan-Ziegler's work [19,20]. Also see [3,4,5,7,9,10] for other recent papers studying strength.…”

“…d , so that we may take V = K d . We may take q (1,1) := x d 1 ∈ S d (V (1,1) ) and q (1,2) (1,2) ), where x 1 , . .…”

Cubics in 10 variables vs. cubics in 1000 variables: Uniformity phenomena for bounded degree polynomials

“…, d r ) from Stillman's Conjecture (see Conjecture 7.4), but the bound was not worked out explicitly in [2]. However, in their very recent paper [3], Ananyan and Hochster provide explicit such bounds for ideals generated in degree at most 4. By contrast, the methods of [26] and [20] are inherently ineffective.…”