Topology of mixed hypersurfaces of cyclic type (Q1743707)

A polynomial \(f( z,{\overline z})=\sum_{i=1}^m c_il z^{\nu_i}{\overline z}^{\mu_i}\) where \(c_i\in\mathbb C^*\) and \(z^{\nu_i}=z_i^{\nu_{i,1}}\cdots z_n^{\nu_{i,n}}\), \({\overline z}^{\mu_i}=\overline z_1^{\mu_{i,1}}\cdots \overline z_n^{\mu_{i,n}}\) is called a mixed polynomial. A point \(w\in \mathbb C^n\) is called a mixed singular point of \(f(z,{\overline z})\) if \(\mathfrak R f\) and \(\mathfrak I f\) are linearly dependent at \(w\). \textit{M.\ Oka} [Kodai Math.\ J.\ 33, 1--62 (2010; Zbl 1195.14061)] showed the existence of Milnor fibration for the class of strongly non-degenerate mixed polynomials. Let \(p_1,\ldots,p_n\) and \(q_1,\ldots,q_n\) be integers with \(\gcd(p_1,\ldots,p_n)=\gcd(q_1,\ldots,q_n)=1\). Now \(S^1\) and \(\mathbb R^*\) act on \(\mathbb C^n\) as follows: \(c\circ z=(c^{p_1}z_1,\ldots,c^{p_n}z_n)\) where \(c\in S^1\), \(r\circ z=(r^{q_1}z_1,\ldots,r^{q_n}z_n)\) where \(r\in \mathbb R^*\). The mixed polynomial \(f( z,{\overline z})\) is said to be a polar weighted resp.\ a radial weighted homogeneous polynomial if there exists a positive integer \(d_p\) such that \(f(c^{p_1}z_1,\ldots,c^{p_n}z_n,\overline c^{p_1}\overline z_1,\ldots, \overline c^{p_n}\overline z_n)=c^{d_p}f( z,{\overline z})\) for \(c\in S^1\) resp.\ if there exists a positive integer \(d_r\) such that \(f(r^{q_1}z_1,\ldots,r^{q_n}z_n, r^{q_1}\overline z_1,\ldots, r^{q_n}\overline z_n)=r^{d_r}f(z,{\overline z})\) for \(r\in\mathbb R^*\). This class of mixed polynomials was introduced by \textit{M.\ A.\ S.\ Ruas} et al. [in: Trends in Mathematics. Basel: Birkhäuser. 191--213 (2003; Zbl 1172.32008)] and \textit{J.\ L.\ Cisneros-Molina} [Contemp.\ Math.\ 475, 43--59 (2008; Zbl 1172.32008)]; they showed that a polar and radial weighted polynomial \(f\) admits the global Milnor fibration \(f: \mathbb C^n\setminus f^{-1}(0)\to \mathbb C^*\). For a mixed polynomial \(f(z,{\overline z})=\sum_{i=1}^m c_iz^{\nu_i}{\overline z}^{\mu_i}\) with \(c_i\neq 0\) for \(i\in\{1,\ldots,m\}\) set \(g(z)=\sum_{i=1}^m c_i z^{\nu_i-\mu_i}\); \(g\) is called the associated Laurent polynomial of \(f\). \textit{M.\ Oka} [Contemp.\ Math.\ 538, 389--399 (2011; Zbl 1214.32008)] studied the following polar weighted homogeneous polynomials \[ \begin{aligned} f_{a, b}( z, {\overline z})&=z_1^{a_1+b_1}\overline z_1^{b_1}+\cdots+z_{n-1}^{a_{n-1}+b_{n-1}}\overline z_{n-1}^{b_{n-1}}+z_n^{a_n+b_n}\overline z_n^{b_n}, \\ f_I(z,{\overline z})&=z_1^{a_1+b_1}\overline z_1^{b_1}z_2+\cdots+z_{n-1}^{a_{n-1}+b_{n-1}}\overline z_{n-1}^{b_{n-1}}z_n+z_n^{a_n+b_n}\overline z_n^{b_n},\\ f_{II}(z,{\overline z})&=z_1^{a_1+b_1}\overline z_1^{b_1}z_2+\cdots+z_{n-1}^{a_{n-1}+b_{n-1}}\overline z_{n-1}^{b_{n-1}}z_n+z_n^{a_n+b_n}\overline z_n^{b_n}z_1, \end{aligned} \] where \(a_j\geq 1\) and \(b_j\geq 0\) for \(j\in\{1,\ldots,n\}\). He showed that for \(\iota=( a,b)\) and \(\iota=I\) the two links of \(f_\iota\) and the associated Laurent polynomial \(g_\iota(z)\) in a small sphere are isotopic and their Milnor fibrations are isomorphic. He conjectured the assertion will also be true for the case \(f_{II}\). This is shown in this paper: It is a simple consequence of Cor.\ 3 of the main result of this paper, the tranversality theorem 1.