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Statement

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

Question (p. 148, in the note added in proof, unnumbered). "Let the restriction that $|z_\nu|\leq1$ be removed, and for $c>0$ let $N_n(c)$ denote the supremum of the number of components of $E$ whose diameter is greater than $1+c$. Is the sequence $\{N_n(c)\}$ bounded, for each fixed value of $c$?"

It follows the note's answer to Problem 9, which shows that for the threshold $1$ (and components of $\bar E$) the count grows at least like $n/2$. The question counts components of $E$, not of $\bar E$.

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

Read depth. Claims checked: the question was read clause by clause on the page image of p. 148 on 2026-10-08. Nothing here is independently reviewed.

Dependencies

Problem 9 and its answer.

Bears on

• #511: the problem is this question with the threshold written $c>1$ in place of the paper's $1+c$, $c>0$; both count components of the open set $\{|f|<1\}$ with no restriction on the zeros.