Abstract
Let P be a lattice polytope with the \(h^{*}\)-vector \((1, h^*_1, \ldots , h^*_s)\). In this note we show that if \(h_s^* \le h_1^*\), then the Ehrhart ring \({\mathbb {k}}[P]\) is generated in degrees at most \(s-1\) as a \({\mathbb {k}}\)-algebra. In particular, if \(s=2\) and \(h_2^* \le h_1^*\), then P is IDP. To see this, we show the corresponding statement for semi-standard graded Cohen–Macaulay domains over algebraically closed fields.
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Acknowledgements
The authors thank Kazuma Shimomoto for many inspiring discussions throughout this project. They also thank anonymous reviewers for their careful reading and valuable comments.
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This work was partially supported by JSPS Grant-in-Aid for Scientific Research (C) 19K03456.
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Katthän, L., Yanagawa, K. Graded Cohen–Macaulay Domains and Lattice Polytopes with Short h-Vector. Discrete Comput Geom 68, 608–617 (2022). https://doi.org/10.1007/s00454-021-00342-z
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DOI: https://doi.org/10.1007/s00454-021-00342-z