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.

中文翻译：

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分级 Cohen-Macaulay 域和具有短 h 向量的晶格多胞体

令P是具有\(h^{*}\) -vector \((1, h^*_1, \ldots , h^*_s)\)的晶格多胞体。在本笔记中，我们表明如果\(h_s^* \le h_1^*\)，那么 Ehrhart 环\({\mathbb {k}}[P]\)最多以度数生成\(s-1\ )作为\({\mathbb {k}}\) -代数。特别是，如果\(s=2\)和\(h_2^* \le h_1^*\)，则P是 IDP。为了看到这一点，我们展示了代数闭域上半标准分级 Cohen-Macaulay 域的相应陈述。