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arXiv · 1711.04206

Rationality of Poincaré Series for a Family of Lattice Simplices

Benjamin Braun·Brian Davis
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Abstract

We investigate multi-graded Gorenstein semigroup algebras associated with an infinite family of reflexive lattice simplices. For each of these algebras, we prove that their multigraded Poincaré series is rational. Our method of proof is to produce for each algebra an explicit minimal free resolution of the ground field, in which the resolution reflects the recursive structure encoded in the denominator of the finely-graded Poincaré series. Using this resolution, we show that these algebras are not Koszul, and therefore rationality is non-trivial. Our results demonstrate how interactions between multivariate and univariate rational generating functions can create subtle complications when attempting to use rational Poincaré series to inform the construction of minimal resolutions.

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Benjamin Braun, Brian Davis. 2020-10-30. Rationality of Poincaré Series for a Family of Lattice Simplices. https://arxiv.org/abs/1711.04206

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