Define a polynomial $f: \mathbb{C}^2 \to \mathbb{C}$ by $f(x,y)= x(x(2y+1)+1)(x(2y+1)-1).$ The inverse image of zero (i.e. $f^{-1}(0)$) is $\mathbb{C}\cup \mathbb{C}^*\cup \mathbb{C}^*$ (the unions are disjoint).
What is the homotopy type of the general fiber $f^{-1}(c)$, where c is not zero?
What is the bifurcation set for this polynomial?
I have tried to solve this by using Nemethi-Sibastiani-Thom's theorem but could not solve it.
Thanks in advance!!
