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Statement

Notation (p. 125): $f$ is a monic polynomial (1) of degree $n$ and $E$ the set where $|f|<1$.

Problem 6 (p. 139). "Suppose that $F$ is a closed set of transfinite diameter 1, and that $F$ is not contained in any closed disk of radius 1. Can the number of components of $E$ (all $z_\nu\in F$) be $n$? If the transfinite diameter of $F$ is less than 1, does there exist a positive constant $c$ (depending on $F$, or perhaps only on the transfinite diameter of $F$) such that $E$ has at most $(1-c)n$ components, when $n$ is large?"

Source. P. Erdős, F. Herzog, G. Piranian, Metric properties of polynomials, J. Analyse Math. 6 (1958), 125--148, doi:10.1007/BF02790232; Problem 6 on p. 139. 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. 139 on 2026-10-08. Nothing here is independently reviewed.

Dependencies

Section 5's discussion (p. 136): for zeros in the closed unit disk $E$ can have $n$ components ($z^n+1$), and Theorem 7 on the components of $\bar E$.

Bears on

• #1042: the problem's two questions are those of Problem 6; the paper's second question carries the qualifier "when $n$ is large", which the problem's statement omits; the problem page's Formulation note reads the question with it.