# Intersection of superlevel set of polynomials

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!

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.)
• I didn’t understand your comment in the bracket because $|P_3|$ need not be harmonic. Also can we at least find polynomial $P_3$ whose superlevel set is close (may be in Housdorff metric sense) to intersection of superlevel sets of $P_1$ and $P_2$? Apr 19 '19 at 11:50
• @Mayuresh: $\log|P|$ is harmonic (away from the zeros of $P$) and has the same level lines. Apr 19 '19 at 16:27