Local minima of the modulus of a square-free complex polynomial is always zero?

Question

I came across the following questions in my research:

Let P(z) be a complex polynomials which has only simple roots. Consdier the two real-variable function h(x,y)=|P(x+yi)|. Then is it true that h has no positive local minima?

If P(x) is real polynomial, then it's not very difficult to see that it is true. It's a straightforward application of Rolle's Theorem

Answer 1

Suppose that there is $(x_0,y_0) \in \mathbb R^2$ and an open neigborhood $U$ of $(x_0,y_0)$ such that

$$0<h(x_0,y_0) \le h(x,y)$$

for all $(x,y) \in U$. With $z_0=x_0+iy_0$ we have

$$0<|P(z_0)| \le |P(z)|$$

for all $z \in U.$ Since $P$ is holomorphic on $U$, the minimum principle shows that $P$ is constant on $U$. The identity theorem shows then that $P$ is constant on $ \mathbb C$, hence $h$ is constant on $ \mathbb R^2.$