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Statement

Notation (p. 125): $E(f)$ is the set where $|f|<1$ and $\bar E(f)$ its closure.

Theorem 7 (p. 136). "If $n$ is sufficiently large and

$$P_n(z)=\frac{(z^n+1)(z-1)^2}{(z-e^{i\pi/n})(z-e^{-i\pi/n})},$$

then the set $\bar E(P_n)$ has $n-1$ components."

$P_n$ is monic of degree $n$ with all zeros on the unit circle: the zeros of $z^n+1$ other than $e^{\pm i\pi/n}$, and a double zero at $1$. The paper sets the theorem against two facts it cites (p. 136): for zeros in $\bar D$ the open set $E$ can have $n$ components ($z^n+1$), while $\bar E$ has at most $n-1$ components (Szegő, its [8]). Theorem 7 shows that this bound "can not be improved, at least when $n$ is sufficiently large" (p. 136). The same section notes that for zeros on $I=[-1,1]$, Theorem 1 caps the number of components of $E$ at $1+[n/2]$.

Source. P. Erdős, F. Herzog, G. Piranian, Metric properties of polynomials, J. Analyse Math. 6 (1958), 125--148, doi:10.1007/BF02790232; Theorem 7 on p. 136, its proof with Figure 1 on pp. 136--139. The copy read is identified on the source card.

Read depth. Claims checked: the statement and the paragraph before it were read on the page image of p. 136 on 2026-10-08; the proof was read for structure, not checked. Nothing here is independently reviewed.

Proof pointer

Pages 136--139. The plane is cut into $n-1$ regions, each holding one zero of $P_n$, by the $n-1$ rays $\arg z=2\pi\nu/n$ (shortened in the right half-plane) and two arcs near $z=1$: $r=\cos\vartheta$ for $2n^{-1/3}\le|\vartheta|\le\pi/2$ and $r=1-|\vartheta|/2$ for $2\pi/n\le|\vartheta|\le2n^{-1/3}+2\pi/n$. Writing $P_n=(z^n+1)/Q_n$, the proof shows $|Q_n|<|z^n+1|$, that is $|P_n|>1$, on every cut except at the origin, through the explicit formula (5) for $|Q_n(re^{i\vartheta})|^2$ and separate estimates on the left-half-plane rays, the two arcs and the right-half-plane rays. Since $P_n'(0)\ne0$, the origin is not a multiple point of $|P_n|=1$, and $\bar E(P_n)$ has $n-1$ components.

Dependencies

Szegő's bound (the paper's [8]) for the statement that the result is sharp; the proof itself uses nothing else from the paper.

Bears on

• #1042: the components of $E$ for zeros in a closed set $F$. Here $F$ is the unit circle, of transfinite diameter $1$ but inside a closed disk of radius $1$, the case that Problem 6 excludes; the theorem counts components of $\bar E$, not of $E$.