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Source. A. W. Goodman, On the convexity of the level curves of a polynomial, Proc. Amer. Math. Soc. 17 (1966), no. 2, 358--361, DOI 10.1090/S0002-9939-1966-0188408-3, identified on the source card: section 4, "Some open questions", p. 361, with the setting on p. 358.

Read depth. Claims checked: the passage was read clause by clause on the page image, and the level $c$ was recomputed here as the common value of $|P|$ at the critical points; the numerical check under the proof pointer was also made here. The double points and their position are taken as printed; the paper gives no proof. Nothing here is independently reviewed.

Statement

With the paper's notation (p. 358), $E(c)=\{z:|P(z)|<c\}$ is open and its boundary $\Gamma(c)$ is the lemniscate $|P(z)|=c$.

Example (p. 361, suggested by the referee). Let $P(z)=z(z^5-1)$ and $c=5/6^{6/5}$. Then $\Gamma(c)$ has five double points, at which the curve crosses itself at right angles, and all five lie on the boundary of the component of $E(c)$ containing $z=0$; so that component is not convex.

$P$ has six simple roots, $0$ and the fifth roots of unity. Its critical points are the five roots of $z^5=1/6$, at each of which $|P(z)|=6^{-1/5}\cdot5/6=5/6^{6/5}$, so $c$ is the common critical value. The paper does not state the number of components of $E(c)$.

Proof pointer

P. 361 states the example without proof: the paper's only reason for nonconvexity is that the five double points lie on the boundary of the component containing $0$. The level is the computation above. The argument of the first counterexample would finish it: a convex component contains in its closure the segment between two of its boundary points, and the midpoint of two adjacent critical points, $6^{-1/5}\cos(\pi/5)e^{i\pi/5}$, has $|P|\approx0.598$, above $c\approx0.582$ (a numerical check made here, not in the paper).

Dependencies

None stated in the paper.

Bears on

• Problem 1047: an example bearing on Grunsky's question for the open set $E(c)$ at a critical level. The paper does not count the components of $E(c)$ or treat the closed set $\{z:|P(z)|\le c\}$ of the problem, which at this level contains the five double points; it does not by itself answer the problem as posed.