In *The Shape of Level Curves* (link to article on JSTOR), George Piranian constructs a polynomial $p$ with $n$ distinct zeros, such that the set $\{z:|p(z)|<\epsilon\}$ has $n$ components (each of which contains a zero of course, by the minimum modulus theorem), and such that some of these components are not star-shaped with respect to the zeros they contain.

I notice however that in this construction, the components (also called polynomial tracts) which are not star-shaped with respect to their zeros, are exactly the ones such that the zeros contained therein have very high multiplicity. Thus my question: **If a polynomial tract (say a component of $\{z:|p(z)|<\epsilon\}$) contains a single simple zero of $p$, will it be starshaped with respect to that zero?**

I am not sure whether the assumption that there are $n$ components ($n$ equaling the number of distinct zeros of $p$) of $\{z:|p(z)|<\epsilon\}$ plays a role or not.