4 Corrected a serious mathematical error in the last paragraph

Now suppose that the roots of $P$ lie on a line. We can assume that $P$ is real (with positive leading term) and all of the roots of $P$ are real as well. Let $\beta$ and $\gamma$ be the least and greatest root of $P$, respectively. Then, by a similar analysis, if $\mathrm{Hull}(P) = \mathrm{Conv}(P)$, then either the degree of $P$ is even, mathrm{Conv}(P)$if and some only if •$\Pi_\omega$satisfies Bégassat's sufficient condition \Pi_y$ has all real roots for some value of $\mathrm{Hull}(P) = \mathrm{Conv}(P)$ (described in his comment numbered 3), or the degree y$, • The roots of$P$is odd, and \Pi_{\Pi(\beta)}$ have real part at least $\Pi(\mu_1) = \Pi(\mu_n)$, \beta$, and • The roots of$\Pi(\mu_{j}) \le \Pi(\mu_1)$when \Pi_{\Pi(\gamma)}$ have real part at most $j$ is odd \gamma$. • You can easily verify that 1, 2, and$\Pi(\mu_{j}) \ge \Pi(\mu_1)$3 hold when$j$is evenP$ has degree 3. I would be tempted to conjecture that 2 and 3 always hold (Here $\mu_1, \ldots, \mu_n$ are when the roots of $P$). Note that in the latter case, $P$ satisfies the $A^2B$ criterion given aboveare real).

If

We can characterize those $P$ for which $\mathrm{Hull}(P) = \mathrm{Conv}(P)$.

First suppose that the roots of $P$ do not lie on a line. We prove that $\mathrm{Hull}(P) = \mathrm{Conv}(P)$ if and only if there is an antiderivative $Q$ of $P$ for which $Q = A^2B$, and the roots of $B$ lie in the convex hull of $A$, then $\mathrm{Conv}(Q) = \mathrm{Conv}(A) = \mathrm{Conv}(P)$, so $\mathrm{Hull}(P) = \mathrm{Conv}(P)$. We prove that the converse holds whenever $\mathrm{Conv}(P)$ A$. The "if" is not a line segmentimmediate and left to the reader. Note in this case that a root$\beta$lies in of$P$can lie on the interior boundary of$\mathrm{Conv}(\Pi_\omega)$whenever only if$\beta$is not a root of$\Pi_\omega$. \Pi_\omega$, or all the roots of $\Pi_\omega$ lie on a line. This latter case of course is impossible if the roots of $P$ do not lie on a line.

A similar analysis shows

Now suppose that if the roots of $P$ lie on a line. We can assume that $P$ is real (with positive leading term), term) and all of the roots of $P$ are real as well. Then, and by a similar analysis, if $\mathrm{Hull}(P) = \mathrm{Conv}(P)$, then either the degree of $P$ is even, and some $\Pi_\omega$ satisfies Bégassat's sufficient condition for $\mathrm{Hull}(P) = \mathrm{Conv}(P)$ (described in his comment numbered 3), or the degree of $P$ is odd, and $\Pi(\mu_1) = \Pi(\mu_n)$, and $\Pi(\mu_{j}) \le \Pi(\mu_1)$ when $j$ is odd and $\Pi(\mu_{j}) \ge \Pi(\mu_1)$ when $j$ is even. (Here $\mu_1, \ldots, \mu_n$ are the roots of $P$). Note that in the latter case, $P$ satisfies the $A^2B$ criterion given above.

2 deleted 5 characters in body

If there is an antiderivative $Q$ of $P$ for which $Q = A^2B$, and the roots of $B$ lie in the convex hull of $A$, then $\mathrm{Conv}(Q) = \mathrm{Conv}(A) = \mathrm{Conv}(P)$, so $\mathrm{Hull}(P) = \mathrm{Conv}(P)$. We prove that the converse holds whenever $\mathrm{Conv}(P)$ is not a line segment. Note that in this case that a root $\beta$ lies in the interior of $\mathrm{Conv}(\Pi_\omega)$ whenever $\beta$ is not a root of $\Pi_\omega$.

Let $\beta$ be a root of $P$. We claim that for every neighborhood $N$ of $\Pi(\beta)$ there is a neighborhood $M$ of $\beta$ such that $\mathrm{Conv}(\Pi_y) \supset M$ for every $y \in \mathbb C \setminus N$. This certainly holds for any given $M$ when $y$ lies outside a large compact subset of $\mathbb C$, so we can think of $y$ ranging over a compact set. For each $y \neq \Pi(\beta)$, there is a ball of positive radius around $\beta$ in $\mathrm{Conv}(\Pi_y)$, and the size of the maximal such ball varies continuously, so the claim follows.

Now, suppose that $\beta$ and $\gamma$ are adjacent vertices (extreme points) of $\mathrm{Conv}(P)$. Suppose $\gamma$ is not a root of $\Pi_{\Pi(\beta)}$. Then $\gamma$ lies in the interior of $\mathrm{Conv}(\Pi_{\Pi(\beta)})$, and we can find $\gamma'$ in the interior of $\mathrm{Conv}(\Pi_{\Pi(\beta)})$ so that $\overline{\gamma' \beta} \cap \mathrm{Conv}(P) = \beta$. Then $\overline{\gamma' \beta} \subset \mathrm{Conv}(\Pi_y)$ for all $y$ in a suitable neighborhood $N$ of $\Pi(\beta)$. On the other hand, $M \subset \mathrm{Conv}(\Pi_y)$ for a suitable neighborhood $M$ of $\beta$ and all $y \notin N$. Therefore $\overline{\gamma' \beta} \cap M \subset \mathrm{Hull}(P)$, and hence $\mathrm{Conv}(P) \subsetneq \mathrm{Hull}(P)$.

So if $\mathrm{Conv}(P) = \mathrm{Hull}(P)$, then $\Pi(\beta) = \Pi(\gamma)$ for all adjacent vertices $\beta, \gamma$ of $\mathrm{Conv}(P)$. Letting $Q$ be $\Pi(\beta)$ for any extreme point $\beta$ of $\mathrm{Conv}(P)$, we see that every extreme point of $\mathrm{Conv}(P)$ is a root of $Q$, and hence a double root of $Q$. Moveover, if $\mathrm{Conv}(P) = \mathrm{Hull}(P)$, then $\mathrm{Conv}(P) \supseteq \mathrm{Conv}(Q)$, so $Q = A^2B$, where the roots of $B$ lie in the convex hull of the roots of $A$, the extreme points of $\mathrm{Conv}(P)$.

A similar analysis shows that if $P$ is real (with positive leading term), and all of the roots of $P$ are real, and $\mathrm{Hull}(P) = \mathrm{Conv}(P)$, then either the degree of $P$ is even, and some $\Pi_\omega$ satisfies Bégassat's sufficient condition for $\mathrm{Hull}(P) = \mathrm{Conv}(P)$ (described in his comment numbered 3), or the degree of $P$ is odd, and $\Pi(\mu_1) = \Pi(\mu_n)$, and $\Pi(\mu_{j}) \le \Pi(\mu_1)$ when $j$ is odd and $\Pi(\mu_{j}) \ge \Pi(\mu_1)$ when $j$ is even. (Here $\mu_1, \ldots, \mu_n$ are the roots of $P$). Note that in the latter case, $P$ satisfies the $A^2B$ criterion given above.