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Statement

Problem 16 (p. 145). "Let $\{z_\mu\}_{\mu=1}^m$ be a set of distinct points, and let

$$f(z)=\prod_{\mu=1}^m(z-z_\mu)^{k_\mu},$$

where the $k_\mu$ are positive integers. Let $c$ be a constant small enough so that the lemniscate $|f(z)|=c$ consists of $m$ distinct loops. Are all the loops convex?"

The paper credits the question to H. Grunsky (private communication) and calls it related to Theorem 11 (p. 145).

Source. P. Erdős, F. Herzog, G. Piranian, Metric properties of polynomials, J. Analyse Math. 6 (1958), 125--148, doi:10.1007/BF02790232; Problem 16 on p. 145. The copy read is identified on the source card.

Read depth. Claims checked: the problem was read clause by clause on the page image of p. 145 on 2026-10-08. Nothing here is independently reviewed.

Dependencies

None.

Bears on

• #1047: the problem asks Problem 16 for the $m$ components of the closed set $\{|f|\le c\}$ of a monic polynomial with $m$ distinct roots; the paper asks about the $m$ loops of the curve $|f|=c$.