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

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Let K be a field and let S = K[x₁,..., xₙ] be a polynomial ring. Consider a homogenous ideal I in S. Let tᵢ denote reg(Torᵢ (S/I, K)), the maximal degree of an ith syzygy of S/I. We prove bounds on the numbers tᵢ for i > n/2 purely in terms of the previous tᵢ. As a result, we give bounds on the regularity of S/I in terms of as few as half of the numbers tᵢ. 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 t₁.