Two different ways of calculating critical points are given from the formulas (I hand wrote them at a class so it's not 100% the formulas but they're close):
$(-1)^{dim} \mathcal{X}(X)$ = # of critical points of the product $\pi_{i=1}^{X}2i^{ui}|_{X}$ for general $u_{I}$
And
$(-1)^{dim{X}}\mathcal{X}(X \backslash H)$ for general $a \in \mathbb{C}$.
For example $X={x^{2}+y^{2}=1}\in \mathbb{C}^{2}$
$Ax+By|_{X}$ = # of critical points = 2
In the example a line intersects a circle and one of the formulas probably yields the number of critical points, how is the example done?
I know $\mathcal{X}$ is the Euler characteristic but I don't know which formula to use or how I can count vertices of the line segment without the point intersected with the circle.
EDIT: I think the # of critical points is the function of x and y, Ax+By, restricted to the domain of the circle. Which formula is it?