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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: the setting on p. 358, the construction in section 3 (pp. 359--361) and the unnumbered Theorem on p. 361.

Read depth. Claims checked: the setting and the Theorem were read clause by clause on the page images. In the construction, the value $c^2=|Q(i)|^2$ and the algebra turning the conditions on $Q(a)$, $|Q(a)|$ and $|Q(0)|$ into (7), (8) and (9) were recomputed here, as was the check that $a=5$, $b=303$ satisfy all three; the topological steps (where the roots lie, the count of components, and the nonconvexity of the component of the smaller positive root, which the paper argues only by pointing to section 2) were read but not checked. Nothing here is independently reviewed.

Statement

Setting (p. 358). For $m$ distinct points $z_1,\dots,z_m$ and positive integers $k_\alpha$, equation (1) is $P(z)=\prod_{\alpha=1}^m(z-z_\alpha)^{k_\alpha}$. $E(c)$ is the open set of $z$ with $|P(z)|<c$, and $\Gamma(c)$, its boundary, is the lemniscate $|P(z)|=c$. Grunsky's question (problem 16 of Erdős, Herzog and Piranian, as the paper reports it): if $E(c)$ has $m$ components, is each of them convex?

Theorem (p. 361, quoted). "Let $c^2=91{,}600$ and let (10) $Q(z)=3z^4-20z^3+6z^2-60z+303$. Then the polynomial $Q(z)$ has four distinct roots and one of the components of $E(c)$ is not convex."

Here $E(c)=\{z:|Q(z)|<c\}$ with $c=\sqrt{91{,}600}$. The paper adds that dividing $Q$ and $c$ by $3$ gives the monic form of (1); $E(c)$ is unchanged. $Q$ is the case $a=5$, $b=303$ of the family (6) below, for which the section's argument gives four components of $E(c)$, one about each root, so the Theorem answers Grunsky's question no with all roots simple ($m$ equal to the degree, $4$).

The family behind it (pp. 359--360). For $a>0$ and $b>0$ the paper takes $Q'(z)=12(z^2+1)(z-a)$ (5), so $Q(z)=3z^4-4az^3+6z^2-12az+b$ (6), and sets $c=|Q(i)|$, a critical value, with $c^2=(b-3)^2+64a^2$. It states three conditions:

• (7) $0<b<a^2(a^2+6)$, equivalent to $Q(a)<0$: then $Q$ has four distinct roots, a conjugate pair $z_1,z_2$ with negative real part and two real roots $0<z_3<a<z_4$.
• (8) $(a^4+6a^2-3)(a^4+6a^2-2b+3)>64a^2$: assuming it, $E(c)$ has four components $E_1,\dots,E_4$ with $z_\alpha\in E_\alpha$. The paper derives it from $|Q(a)|\ge c$, written as $(a^2(a^2+6)-b)^2\ge(b-3)^2+64a^2$, whose rearranged form would carry $\ge$; the printed (8) is strict.
• (9) $6b>9+64a^2$, equivalent to $|Q(0)|>c$: then $E_3$ is not convex.

Proof pointer

Pp. 359--361. The construction splits the double root $2$ of the first counterexample (example_p359) into two simple real roots by prescribing the critical points $\pm i$ and $a$ rather than the roots. $Q$ has no negative real roots, since every term of (6) is positive at a negative real $z$; with the Gauss--Lucas theorem this places a conjugate pair of roots in the left half plane, and $Q(0)=b>0$ together with $Q(a)<0$ places one real root on each side of $a$. Taking $c$ at the critical value $|Q(\pm i)|$, the components of $z_1$ and $z_2$ are separate, and by the symmetry of $E(c)$ in the real axis the rest splits into two components exactly when $|Q(a)|\ge c$. For $E_3$ the paper says only that, following the pattern of section 2, it is not convex if $|Q(0)|>c$; there the double points at the critical points lay on the boundary of the nonconvex component and their midpoint lay outside it, and here $0$ is the midpoint of the critical points $\pm i$. The choice $a=5$, $b=303$ satisfies (7)--(9): $303<25\cdot31=775$, $772\cdot172=132{,}784>1{,}600$, and $1{,}818>1{,}609$; then $c^2=300^2+1{,}600=91{,}600$.

Dependencies

The Gauss--Lucas theorem; otherwise elementary. The construction follows the pattern of the first counterexample (p. 359).

Bears on

• Problem 1047: the Theorem answers Grunsky's question no for the open set $E(c)$ with a quartic whose four roots are simple. The problem asks about the closed set $\{z:|f(z)|\le c\}$. At the paper's level $c=|Q(i)|$ the critical points $\pm i$ of $Q$ lie on the lemniscate, and the paper counts components of the open set only; it does not treat the closed set at that level or at a lower one, so the Theorem does not by itself answer the problem as posed.