A Polynomial Bound on the Regularity of an Ideal in Terms of Half of the Syzygies

TL;DR

This paper establishes polynomial bounds on the regularity of ideals in polynomial rings based on half of their syzygies, improving upon previous doubly exponential bounds.

Contribution

It introduces new polynomial bounds on the regularity of ideals using only half of the syzygies, advancing understanding of syzygy-based regularity bounds.

Findings

Bounds on t_i for i > n/2 in terms of previous t_i

Regularity bounds based on fewer syzygies

Bounds are often much smaller than doubly exponential bounds

Abstract

Let K be a field and let S = K[x_1, ..., x_n] be a polynomial ring. Consider a homogenous ideal I in S. Let t_i denote reg(Tor_i (S/I, K)), the maximal degree of an ith syzygy of S/I. We prove bounds on the numbers t_i for i > n/2 purely in terms of the previous t_i. As a result, we give bounds on the regularity of S/I in terms of as few as half of the numbers t_i. We also prove related bounds for arbitrary modules. These bounds are often much smaller than the known doubly exponential bound on regularity purely in terms of t_1.