Log-concavity of asymptotic multigraded Hilbert series

McCabe, Adam; Smith, Gregory

doi:10.1090/S0002-9939-2012-11808-8

Log-concavity of asymptotic multigraded Hilbert series
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by Adam McCabe and Gregory G. Smith

Proc. Amer. Math. Soc. 141 (2013), 1883-1892

DOI: https://doi.org/10.1090/S0002-9939-2012-11808-8

Published electronically: December 20, 2012

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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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