Let $P\_1$ and $P\_2$ be complex polynomials with complex coefficients and $c > 0$. Can we find polynomial $P\_3$ and $c’>0$ such that



> 
> $\{z \in \mathbb C : |P\_1(z)| \geq c\} \cap \{ z \in \mathbb C : |P\_2(z)| \geq c\}= \{ z \in \mathbb C : |P\_3(z)| \geq c’\}$
> 
> 
> 


holds?


It seems like this is either trivially false or very hard problem. I don’t have much idea. Any suggestion or reference is welcome. Thanks!



Intersection of superlevel set of polynomials
You are right: this is trivially false.


Let $P\_1(z)=z^2,\; P\_2(z)=(z-a)^2$, and $c=1$. Then the boundaries of the first two level sets are circles of radius $1$,
and choosing an appropriate $a$ you can make them cross at any given angle.


On the other hand the boundary of the set in the RHS is a polynomial lemniscate,
and it is clear that it cannot cross itself under any angle except $\pi/n$ where
$n$ is an integer. (This is a general local property of level sets of harmonic functions,
which is easy to prove.)


