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

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