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arXiv · 1109.4135

Log-concavity of asymptotic multigraded Hilbert series

Adam McCabe·Gregory G. Smith
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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. In particular, we extend some results by Beck-Stapledon and Diaconis-Fulman beyond the standard graded situation.

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BibTeXRIS

Adam McCabe, Gregory G. Smith. 2011-09-19. Log-concavity of asymptotic multigraded Hilbert series. https://doi.org/10.1090/s0002-9939-2012-11808-8

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