Mathematical Research Letters

Volume 15 (2008)

Number 2

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

Pages: 331 – 349



Juan Migliore (University of Notre Dame)

Uwe Nagel (University of Kentucky)

Fabrizio Zanello (Michigan Technological University)


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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