Goodman 1966 convexity level curves polynomial

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example_p359: Goodman's first counterexample: for P(z) = (z^2 + 1)(z - 2)^2 and c = 5 sqrt(5)/4, the open set where |P(z)| < c has three components and the one containing 2 is not convex; a negative answer to Grunsky's question for the open set.

example_p361: The referee's example reported by Goodman: for P(z) = z(z^5 - 1) and c = 5/6^{6/5}, the lemniscate has five double points, all on the boundary of the component of the open set where |P(z)| < c that contains 0, so that component is not convex.

theorem: Goodman's Theorem: for Q(z) = 3z^4 - 20z^3 + 6z^2 - 60z + 303 and c^2 = 91,600, Q has four distinct roots and some component of the open sublevel set E(c) is not convex; a negative answer to Grunsky's question for the open set at a critical level, not by itself Problem 1047's closed set.

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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 (volume, issue and DOI from the Crossref record; the page heads show only "1966" and "April"). Presented to the Society January 24, 1966; received by the editors August 12, 1965.

The copy read for this card is a publisher scan of the four printed pages 358--361 with a machine text layer (scan p. $n$ is printed p. $357+n$), 365,640 bytes. The text layer garbles the formulas, so the statements below were checked on the page images of pp. 359 and 361. Provenance: the survey download set of September 2026; the download URL was not recorded. No notice is printed on any of the four pages; the publisher's article page could not be read on 2026-10-02 (https://pubs.ams.org/journals/proc/1966-017-02/S0002-9939-1966-0188408-3, reached through the DOI, rendered only its navigation), and the publisher's copyright policy page (https://www.ams.org/publications/authors/ctp, read 2026-10-07) states that the "AMS permits the noncommercial use of its copyrighted works for educational purposes only, such as to quote brief passages or to copy small portions of content for personal use in teaching or research" and names Creative Commons licenses only for its open-access series and for authors' own postings of an accepted manuscript or draft, neither of which covers this publisher scan, every other right reserved.

Read status: claims checked. The Theorem (p. 361) and the first counterexample (p. 359) were read clause by clause on the page images, and the arithmetic of the first counterexample and of the constants $a=5$, $b=303$, $c^2=91{,}600$ in equations (6) and (7)--(9) was recomputed here; the location of the roots and the component count of the second example (pp. 359--361) were not checked.

Contents

Setting (p. 358): $P(z)=\prod_{a=1}^m(z-z_a)^{k_a}$ with $m$ distinct roots, $E(c)=\{z:|P(z)|<c\}$ (an open set) and $\Gamma(c)$ its boundary, the lemniscate $|P(z)|=c$. Grunsky's question, reported as problem 16 of Erdős, Herzog and Piranian (card): if $E(c)$ has $m$ components, is each component convex? The answer is no; Pommerenke had also found a counterexample, of high degree and with a multiple root of high order (p. 358).

• First counterexample (section 2, p. 359; example_p359): $P(z)=(z^2+1)(z-2)^2$, with critical points $z_1^*,z_2^*=(1\pm i)/2$ and $z_3^*=2$, and $c=|P(z_1^*)|=5\sqrt5/4$. At this $c$ the curve $\Gamma(c)$ has double points at $z_1^*$ and $z_2^*$, so $E(c)$ has three components; the component containing $2$ has both $z_1^*$ and $z_2^*$ on its boundary but omits their midpoint $x^*=1/2$, where $|P(1/2)|=45/16>5\sqrt5/4$, so it is not convex.
• Second counterexample and the Theorem (section 3, pp. 359--361; theorem): with $Q'(z)=12(z^2+1)(z-a)$, $a>0$, $Q(z)=3z^4-4az^3+6z^2-12az+b$ (6), $c=|Q(i)|$ and $c^2=(b-3)^2+64a^2$, the conditions are (7) $0<b<a^2(a^2+6)$, under which $Q$ has four distinct roots, two conjugate and two real with $0<z_3<a<z_4$; (8) $(a^4+6a^2-3)(a^4+6a^2-2b+3)>64a^2$, under which $E(c)$ has four components (derived from $|Q(a)|\ge c$, whose rearranged form carries $\ge$, while the printed (8) is strict); and (9) $6b>9+64a^2$, under which the component of $z_3$ omits $0$ and is not convex. They hold for $a=5$, $b=303$. Theorem (p. 361): for $Q(z)=3z^4-20z^3+6z^2-60z+303$, equation (10), and $c^2=91{,}600$, the roots of $Q$ are simple and some component of $E(c)$ fails to be convex. Dividing $Q$ and $c$ by $3$ gives the monic normalization (1).
• Open questions (section 4, p. 361): the maximum number of nonconvex components as a function of the degree; conditions ensuring convexity; the author's conjecture that when $E(c)$ has $m$ components each is starlike with respect to its root, which the referee doubts; and the referee's example $P(z)=z(z^5-1)$ with $c=5/6^{6/5}$, where $\Gamma(c)$ has five double points on the boundary of the component containing $0$ (example_p361).

Compiled scope

The four pages were read on the page images; the statements above were checked and the elementary arithmetic noted under the read status was recomputed, but the root and component-count claims of the second example were not verified, and nothing here is independently reviewed.

Bears on. #1047: the Theorem on p. 361 answers Grunsky's question, posed for the open set $E(c)$, in the negative with a quartic having four simple roots, and the first counterexample (p. 359) does so with $(z^2+1)(z-2)^2$; the referee's example (p. 361) gives a nonconvex component of $E(c)$ for $z(z^5-1)$ without a component count. At the critical levels of the paper's two examples the closed sublevel set of the problem has a different component count, and the paper treats the closed set for none of the three, so none of them by itself answers the problem as posed.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.