Mathematical Research Letters

Volume 13 (2006)

Number 6

An upper estimate of integral points in real simplices with an application to singularity theory

Pages: 911 – 921



Stephen T. Yau (University of Illinois at Chicago)

Letian Zhang (Illinois Mathematics and Science Academy)


Let $\Delta(a_1,a_2,\cdots,a_n)$ be an $n$-dimensional real simplex with vertices at $(a_1,0,\cdots,0), (0,a_2,\cdots,0), \cdots, (0,0,\cdots,a_n)$. Let $P_{(a_1,a_2,\cdots,a_n)}$ be the number of positive integral points lying in $\Delta(a_1,a_2,\cdots,a_n)$. In this paper we prove that $n! P_{(a_1,a_2,\cdots,a_n)}\leq (a_1-1)(a_2-1)\cdots(a_n-1)$. As a consequence we have proved the Durfee conjecture for isolated weighted homogeneous singularities: $n! p_{g} \leq \mu$, where $p_{g}$ and $\mu$ are the geometric genus and Milnor number of the singularity, respectively.

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