Star-shapeness of polynomial tracts containing a single zero

Question

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.

Answer 1

As stated, the answer is negative. I will provide a sketch.

Indeed, consider some non-star-like Jordan domain $V$, and a conformal isomorphism $\phi\colon V\to\mathbb{D}$ from $V$ to the unit disc. Then, for some $r_0$ sufficiently close to $1$, the preimage $V_0$ of the disc $D_0$ of radius $r_0$ around $0$ under $\phi$ is also not star-like.

Now use Runge's theorem to approximate $\phi$ uniformly on a neighbourhood $U$ of $\overline{V_0}$ by a polynomial $p$. If the approximation is sufficiently good, then there will be a unique component of $p^{-1}(D_0)$ contained in $U$, which is also not star-like, and mapped with degree $1$. This completes the proof.

Remark 1. Runge's theorem does not guarantee anything about the other zeros of $p$, so we don't know whether every component of $p^{-1}(D_0)$ contains exactly one zero. Using the technique of quasiconformal folding recently developed by Bishop, I suspect that it is possible to construct an example with this property also, but this uses a lot more technology.

Remark 2. On the other hand, suppose that $\newcommand{\eps}{\varepsilon}\eps_0$ is such that a component $V_0$ of $p^{-1}(\{|z|<\eps_0\})$ contains only a simple zero of $p$, for some polynomial $p$. Then it follows from Koebe's distortion theorem that the component $V$ of $p^{-1}(\{|z|<\eps\})$ is star-like, and even convex, when $\eps<c\eps_0$. Here $c$ is a universal constant.