I recently found out (Piranian, "The Shape of Level Curves") that a polynomial tract (ie a connected component of a set of the form $G=\{z:|p(z)|<\epsilon\}$ for some $\epsilon>0$) need not be starshaped with respect to the zeros of $p$ contained in $G$. This disappointed me bitterly, as that starshapeness was a pivotal step in a proposed "proof" I had of Smale's conjecture.

The places where $G$ is not starshaped with respect to the zeros of $p$ in $G$ are near critical points of $p$ in $G$ or in $\partial G$, so I still hold out a tiny bit of hope for the starshapeness of $G$ with respect to the critical points of $p$ contained in $G$:

Conjecture: If $G$ is a tract of $p$ with smooth boundary containing more than one distinct zero of $p$, then $G$ is starshaped with respect to the critical points of $p$ contained in $G$.

Intuitions/proofs/disproofs/references are all welcome.

EDIT: Note that when I say that $G$ should be "starshaped with respect to the critical points", I mean that each point in $G$ can be seen by some one of the critical points of $p$ in $G$, not of course that some single critical point can see all points in $G$.

Note also that I added the assumption that $G$ contains more than one distinct zero of $p$ (since otherwise $G$ will not contain any critical points of $p$.

One reason I think this is plausible: If we consider the lemniscate of Bernoulli, and let $G$ be the interior of a level curve of $p$ which is a bit bigger, the critical point of $p$ is right in the center, so should be able to "see" both lobes. In the counter-example of Piranian to my desired conjecture (that tracts are star-shaped with respect to the zeros they contain), the points that killed the starshapeness were close to the boundary of $G$, so perhaps if we assume $\partial G$ is smooth, $G$ will contain enough critical points to see into all "corners".

I recently found out (Piranian, "The Shape of Level Curves") that a polynomial tract (ie a connected component of a set of the form $G=\{z:|p(z)|<\epsilon\}$ for some $\epsilon>0$) need not be starshaped with respect to the zeros of $p$ contained in $G$. This disappointed me bitterly, as that starshapeness was a pivotal step in a proposed "proof" I had of Smale's mean value conjecture.

The places where $G$ is not starshaped with respect to the zeros of $p$ in $G$ are near critical points of $p$ in $G$ or in $\partial G$, so I still hold out a tiny bit of hope for the starshapeness of $G$ with respect to the critical points of $p$ contained in $G$:

Conjecture: If $G$ is a tract of $p$ with smooth boundary containing more than one distinct zero of $p$, then $G$ is starshaped with respect to the critical points of $p$ contained in $G$.

Intuitions/proofs/disproofs/references are all welcome.

EDIT: Note that when I say that $G$ should be "starshaped with respect to the critical points", I mean that each point in $G$ can be seen by some one of the critical points of $p$ in $G$, not of course that some single critical point can see all points in $G$.

Note also that I added the assumption that $G$ contains more than one distinct zero of $p$ (since otherwise $G$ will not contain any critical points of $p$.

One reason I think this is plausible: If we consider the lemniscate of Bernoulli, and let $G$ be the interior of a level curve of $p$ which is a bit bigger, the critical point of $p$ is right in the center, so should be able to "see" both lobes. In the counter-example of Piranian to my desired conjecture (that tracts are star-shaped with respect to the zeros they contain), the points that killed the starshapeness were close to the boundary of $G$, so perhaps if we assume $\partial G$ is smooth, $G$ will contain enough critical points to see into all "corners".

I recently found out (Piranian, "The Shape of Level Curves") that a polynomial tract (ie a set of the form $G=\{z:|p(z)|<\epsilon\}$ for some $\epsilon>0$) need not be starshaped with respect to the zeros of $p$ contained in $G$. This disappointed me bitterly, as that starshapeness was a pivotal step in a proposed "proof" I had of Smale's conjecture.

The places where $G$ is not starshaped with respect to the zeros of $p$ in $G$ are near critical points of $p$ in $G$ or in $\partial G$, so I still hold out a tiny bit of hope for the starshapeness of $G$ with respect to the critical points of $p$ contained in $G$:

Conjecture: If $G$ is a tract of $p$ with smooth boundary containing more than one distinct zero of $p$, then $G$ is starshaped with respect to the critical points of $p$ contained in $G$.

Intuitions/proofs/disproofs/references are all welcome.

EDIT: Note that when I say that $G$ should be "starshaped with respect to the critical points", I mean that each point in $G$ can be seen by some one of the critical points of $p$ in $G$, not of course that some single critical point can see all points in $G$.

Note also that I added the assumption that $G$ contains more than one distinct zero of $p$ (since otherwise $G$ will not contain any critical points of $p$.

One reason I think this is plausible: If we consider the lemniscate of Bernoulli, and let $G$ be the interior of a level curve of $p$ which is a bit bigger, the critical point of $p$ is right in the center, so should be able to "see" both lobes. In the counter-example of Piranian to my desired conjecture (that tracts are star-shaped with respect to the zeros they contain), the points that killed the starshapeness were close to the boundary of $G$, so perhaps if we assume $\partial G$ is smooth, $G$ will contain enough critical points to see into all "corners".

I recently found out (Piranian, "The Shape of Level Curves") that a polynomial tract (ie a connected component of a set of the form $G=\{z:|p(z)|<\epsilon\}$ for some $\epsilon>0$) need not be starshaped with respect to the zeros of $p$ contained in $G$. This disappointed me bitterly, as that starshapeness was a pivotal step in a proposed "proof" I had of Smale's conjecture.

The places where $G$ is not starshaped with respect to the zeros of $p$ in $G$ are near critical points of $p$ in $G$ or in $\partial G$, so I still hold out a tiny bit of hope for the starshapeness of $G$ with respect to the critical points of $p$ contained in $G$:

Conjecture: If $G$ is a tract of $p$ with smooth boundary containing more than one distinct zero of $p$, then $G$ is starshaped with respect to the critical points of $p$ contained in $G$.

Intuitions/proofs/disproofs/references are all welcome.

EDIT: Note that when I say that $G$ should be "starshaped with respect to the critical points", I mean that each point in $G$ can be seen by some one of the critical points of $p$ in $G$, not of course that some single critical point can see all points in $G$.

Note also that I added the assumption that $G$ contains more than one distinct zero of $p$ (since otherwise $G$ will not contain any critical points of $p$.

One reason I think this is plausible: If we consider the lemniscate of Bernoulli, and let $G$ be the interior of a level curve of $p$ which is a bit bigger, the critical point of $p$ is right in the center, so should be able to "see" both lobes. In the counter-example of Piranian to my desired conjecture (that tracts are star-shaped with respect to the zeros they contain), the points that killed the starshapeness were close to the boundary of $G$, so perhaps if we assume $\partial G$ is smooth, $G$ will contain enough critical points to see into all "corners".