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

Erman, Daniel; Sam, Steven V; Snowden, Andrew

doi:10.48550/arxiv.1809.09402

Cited by 1 publication

(1 citation statement)

References 44 publications

(77 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Descriptions of earlier work related to this problem [5,6,7,9,15,16,17,28,30] are given in the introductions of [1,2] and in [31]. [12,13,18,19,20,21] contain recent work, some of which utilizes the results of [2], on Stillman's conjecture and related questions, including Noetherianity problems.…”

Section: Introductionmentioning

confidence: 99%

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

Ananyan¹,

Hochster²

2018

Preprint

View full text Add to dashboard Cite

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.

show abstract