TITLE: Number of local maxima of a function 
QUESTION [26 upvotes]: Let $z_j$ ($j=1,\dots, k$) be $k$ points on the complex plane none of which lies on the real line. Is it always true that the function 
$$ F(x)=\sum_{j=1}^k \frac{1}{|x-z_j|^2} $$ has at most $k$ local maxima on the real line?

REPLY [5 votes]: $F(x)=c$ has for any value of $c>0$ at most $2k$ solutions, which in a weak way supports the conjecture that the statement is always true.