Log-concavity of asymptotic multigraded Hilbert series

Adam McCabe, Gregory G. Smith
2012 Proceedings of the American Mathematical Society  
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
more » ... lts by Beck-Stapledon and Diaconis-Fulman beyond the standard graded situation.
doi:10.1090/s0002-9939-2012-11808-8 fatcat:4dxg7r7iuzaufpuajzgbh3layy