多面体的空单形与分次贝蒂数
Empty simplices of polytopes and graded Betti numbers
摘要：
Kalai，Kleinschlaud和Lee关于单纯多胞形的空单形个数的猜想是通过将它与多胞形的第一分次贝蒂数联系起来而建立的。证明使我们能够获得明确的最佳界限的数量空单形的任何给定的尺寸。作为一个关键结果，我们证明了标准分次K-代数的分次贝蒂数关于其Hilbert函数的最优界。
英文摘要：
The conjecture of Kalai, Kleinschmidt, and Lee on the number of empty simplices of a simplicial polytope is established by relating it to the first graded Betti numbers of the polytope. The proof allows us to derive explicit optimal bounds on the number of empty simplices of any given dimension. As a key result, we prove optimal bounds for the graded Betti numbers of any standard graded K-algebra in terms of its Hilbert function.
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