Mathematical Research Letters

Volume 15, Issue 2, March 2008 pp. 331-349.

A characterization of Gorenstein Hilbert functions in codimension four with small initial degree

Authors: Juan Migliore (1), Uwe Nagel (2), and Fabrizio Zanello (3)
Author institution: University of Notre Dame (1), University of Kentucky (2), and Michigan Technological University (3)

Summary: The main goal of this paper is to characterize the Hilbert functions of all (artinian) codimension 4 Gorenstein algebras that have at least two independent relations of degree four. This includes all codimension 4 Gorenstein algebras whose initial relation is of degree at most 3. Our result shows that those Hilbert functions are exactly the so-called {\em SI-sequences} starting with $(1,4,h_2,h_3,...)$, where $h_4 \leq 33$. In particular, these Hilbert functions are all unimodal. We also establish a more general unimodality result, which relies on the values of the Hilbert function not being too big, but is independent of the initial degree.

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