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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 2, "The first counterexample", p. 359, with the setting on p. 358.

Read depth. Claims checked: the example was read clause by clause on the page image, and $P'$, the critical points and the values (3) and (4) were recomputed here. The topological claims (the double points and the count of three components) are taken as printed. 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. 359). Let $P(z)=(z^2+1)(z-2)^2$ (2). Then $P'(z)=2(z-2)(2z^2-2z+1)$, with zeros $z_1^*,z_2^*=(1\pm i)/2$ and $z_3^*=2$. Take $c=|P(z_1^*)|=|P(z_2^*)|=5\sqrt5/4$ (3). At this $c$ the curve $\Gamma(c)$ has double points at $z_1^*$ and $z_2^*$, and $E(c)$ has three components, as many as $P$ has distinct roots ($\pm i$ and $2$). The component $E_3(c)$ containing $2$ is not convex: $z_1^*$ and $z_2^*$ lie on its boundary, but their midpoint $x^*=1/2$ has $|P(1/2)|=45/16$ (4), and $45/16>5\sqrt5/4$, so $x^*$ lies outside the closure of $E_3(c)$.

The root $2$ is double, so $m=3$ is less than the degree $4$; the paper's Theorem (p. 361) gives a quartic with four simple roots.

Proof pointer

P. 359. Everything is direct computation: the factorization of $P'$, the values $|P((1\pm i)/2)|=5\sqrt5/4$ and $|P(1/2)|=45/16$, and the comparison $45/16=2.8125>2.795\ldots=5\sqrt5/4$. If $E_3(c)$ were convex its closure would contain the segment from $z_1^*$ to $z_2^*$, and so $x^*$.

Dependencies

None; elementary computation.

Bears on

• Problem 1047: the example answers Grunsky's question no for the open set $E(c)$ at the critical level $c=5\sqrt5/4$. At that level the closed set $\{z:|P(z)|\le c\}$ of the problem contains the two double points, at which branches of the lemniscate cross, so its components are not those of $E(c)$ and the example as printed does not answer the problem as posed.