Let $f_{ { \bf a},\{ bf b}}({ \bf z},\bar{ \bf z})=z_1^{ a_1+b_1}\bar z_1^{ b_1}+...+z_n^{ a_n+b_n}\bar z_n^{ b_n}$ be a polar weighted homogeneous mixed polynomial with $a_j>0,b_j\ge 0$, $j=1,..., n$ and let $f_{ { \bf a}}({ \bf z})=z_1^{ a_1}+...+z_n^{ a_n}$ be the associated weighted homogeneous polynomial. Consider the corresponding link variety $K_{ { \bf a},{ \bf b}}=f_{ { \bf a},{ \bf b}}\inv(0)\cap S^{ 2n-1}$ and $K_{ { \bf a}}=f_{ { \bf a}}\inv(0)\cap S^{ 2n-1}$. Ruas-Seade-Verjovsky \cite{ R-S-V} proved that the Milnor fibrations of $f_{ { \bf a},{ \bf b}}$ and $f_{ { \bf a}}$ are topologically equivalent and the mixed link $K_{ { \bf a},{ \bf b}}$ is homeomorphic to the complex link $K_{ { \bf a}}$. We will prove that they are $C^\infty$ equivalent and two links are diffeomorphic. We show the same assertion for $ f({ \bf z},\bar{ \bf z})=z_1^{ a_1+b_1}\bar z_1^{ b_1}z_2+...+z_{ n-1}^{ a_{ n-1}+b_{ n-1}}\bar z_{ n-1}^{ b_{ n-1}}z_n+z_n^{ a_n+b_n}\bar z_n^{ b_n}$ and its associated polynomial $ g({ \bf z})=z_1^{ a_1}z_2+...+ z_{ n-1}^{ a_{ n-1}}z_n+z_n^{ a_n}$.
Categories: math.AG