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Stanley's nonunimodal Gorenstein $ h$-vector is optimal


Authors: Juan Migliore and Fabrizio Zanello
Journal: Proc. Amer. Math. Soc. 145 (2017), 1-9
MSC (2010): Primary 13D40; Secondary 13H10, 13E10, 05E40
DOI: https://doi.org/10.1090/proc/13381
Published electronically: September 29, 2016
MathSciNet review: 3565355
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Abstract: We classify all possible $ h$-vectors of graded artinian Gorenstein algebras in socle degree 4 and codimension $ \leq 17$, and in socle degree 5 and codimension $ \leq 25$. We obtain as a consequence that the least number of variables allowing the existence of a nonunimodal Gorenstein $ h$-vector is 13 for socle degree 4, and 17 for socle degree 5. In particular, the smallest nonunimodal Gorenstein $ h$-vector is $ (1,13,12,13,1)$, which was constructed by Stanley in his 1978 seminal paper on level algebras. This solves a longstanding open question in this area. All of our results are characteristic free.


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Additional Information

Juan Migliore
Affiliation: Department of Mathematics, University of Notre Dame, Notre Dame, Indiana 46556
Email: migliore.1@nd.edu

Fabrizio Zanello
Affiliation: Department of Mathematical Sciences, Michigan Technological University, Houghton, Michigan 49931
Email: zanello@mtu.edu

DOI: https://doi.org/10.1090/proc/13381
Keywords: Gorenstein $h$-vector, unimodality, Hilbert function, Macaulay's theorem, $O$-sequence, artinian algebra
Received by editor(s): December 4, 2015
Published electronically: September 29, 2016
Communicated by: Irena Peeva
Article copyright: © Copyright 2016 American Mathematical Society