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Log-concavity of asymptotic multigraded Hilbert series

Authors: Adam McCabe and Gregory G. Smith
Journal: Proc. Amer. Math. Soc. 141 (2013), 1883-1892
MSC (2010): Primary 05E40, 13D40, 52B20
Published electronically: December 20, 2012
MathSciNet review: 3034415
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Abstract: We study the linear map sending the numerator of the rational function representing the Hilbert series of a module to that of its $ r$-th Veronese submodule. We show that the asymptotic behaviour as $ r$ tends to infinity depends on the multidegree of the module and the underlying positively multigraded polynomial ring. More importantly, we give a polyhedral description for the asymptotic polynomial and prove that the coefficients are log-concave.

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Additional Information

Adam McCabe
Affiliation: 35 Summerhill Road, Holland Landing, Ontario, L9N 1C6, Canada

Gregory G. Smith
Affiliation: Department of Mathematics & Statistics, Queen’s University, Kingston, Ontario, K7L 3N6, Canada

Received by editor(s): September 20, 2011
Published electronically: December 20, 2012
Communicated by: Irena Peeva
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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