On the topology of the Newton boundary II
Mutsuo Oka · Journal of the Mathematical Society of Japan · 1980
C^{*}:$ $C-\{0\}$ $z^{ u}=z_{1}^{v_{1}}z_{2}^{ u_{2}}$ ... $z_{n+1}^{ u_{n+1}}$ , $| u|=\sum_{j=1}^{n+1} u_{j}$ for $z\in C^{n+1}$ where $ u=( u_{1}, , u_{n+1})$ .Let $f(z_{1}, \cdots , z_{n+1})=\sum_{ u\in N^{n+1}}c_{ u}z^{ u}$ be an analytic function in a neighborhood of the origin.We denote the Newton boundary of $f$ by $\Gamma(f)$ .(See [7] or [16] for the definition.)$f$ is called to be non-degenerate in the sense of the Newton boundary if $\frac{\partial f_{\Delta}}{\partial z_{1}}=\ldots=\frac{\partial f_{\Delta}}{\partial z_{n+1}}=0$ has no solution in $(C^{*})^{n+1}$ for any closed face $\Delta$ of $\Gamma(f)$ where $f_{\Delta}(z)=\sum_{ u\in\Delta}c_{ u}z^{ u}$ .In this Paper, we use "non-degenerate" in the above sense unless otherwise stated.Now let $f(z)=\sum_{j=1}^{m}c_{j}z^{ u^{j}}$ ($c_{j} eq 0,$ $j=1,$ $\cdots$ , m) be a non-degenerate weighted homogeneous polynomial with the Newton boundary $\Delta$ .Then $\Delta$ is a convex polyhedron spanned by the vertices $ u^{1},$ $\cdots$ , $ u^{m}$ and dim $\Delta=rank\{ u^{1}, \cdots , u^{m}\}-1$ $\leqq n$ .We consider the canonical fibration $f:(C^{*})^{n+1}-f^{-1}(0)\rightarrow C^{*}$ which is the restriction of the Milnor fibration $f:C^{n+1}-f^{-1}(0)\rightarrow C^{*}$ .Let $F^{*}=$ $\{z\in(C^{*})^{n+1} ; f(z)=1\}$ and $F=\{z\in C^{n+1} ; f(z)=1\}$ be the respective fibers and let $\rho$ : $F^{*}\rightarrow(C^{*})^{n+1}$ be the inclusion map.Then $F^{*}$ is completely described by the following theorem.THEOREM (1.1).Assume that dim $\Delta=n$ .Then (i) the Euler-Poincar\'e characteristic $\chi(F^{*})$ is $(-1)^{n}(n+1)$ !$(n+1)$ -volume $(\Delta(0))$ where $\Delta(0)$ is the geometric cone of $\Delta$ and the origin in $R^{n+1}$