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Source. Theorem 1, p. 348, construction pp. 347--348, of P. Erdős, F. Herzog and G. Piranian, Polynomials whose zeros lie on the unit circle, Duke Math. J. 22 (1955), 347--351, DOI 10.1215/S0012-7094-55-02237-7, the edition named on the source card.

Statement

Setting (p. 347, display (1)). $C$ is the unit circle and the polynomials considered are

$$P(z)=\prod_{j=1}^{n}\left(1-\frac{z}{\omega_j}\right),\qquad \lvert\omega_j\rvert=1 ,$$

so $P(0)=1$.

Theorem 1 (p. 348, quoted). "There exists a polynomial (1) such that on every radius of the unit disc there exist points $z'$ and $z''$ with $\lvert P(z')\rvert<1$ and $\lvert P(z'')\rvert>1$."

Context (p. 347). The paper recalls Cohen's theorem that for every such $P$ some path from $0$ to $C$ carries $\lvert P\rvert<1$ everywhere except at $z=0$, and reports, from an oral communication, that C. Loewner had shown that some polynomial (1) exceeds $1$ in modulus somewhere on every radius. Theorem 1 is offered as a very simple explicit example with both properties on every radius.

Read depth. Claims checked: display (1), the statement and the construction of pp. 347--348 were read clause by clause on the page images of the print. The estimates of the construction were followed but not rechecked in detail. Nothing here is independently reviewed.

Proof pointer

Pages 347--348, written here in outline. The example has the shape $P(z)=\prod_{j=1}^{q}\bigl(1+(z/\omega_j)^j\bigr)^{k_j}$ with $\lvert\omega_j\rvert=1$. For each $j$ let $A_j$ (resp. $B_j$) be the set of $\omega\in C$ with $\arg(\omega/\omega_j)^j$ in $[-\pi/3,\pi/3]$ (resp. $[2\pi/3,4\pi/3]$) modulo $2\pi$; each is a union of $j$ disjoint closed arcs of length $2\pi/(3j)$. With radii $r_j=2^{-m(2q-2j+1)}$ and exponents $k_j=2^{m(2q-j)(j-1)}$, the ratios $(k_j/k_p)r_p^{j-p}$ for $j\ne p$ are at most $2^{-m}$, so for large $m$ the $p$-th factor dominates $\log P$ on the circle $\lvert z\rvert=r_p$: there $\lvert P\rvert>1$ in the directions of $A_p$ and $\lvert P\rvert<1$ in those of $B_p$. Since $\sum 1/j$ diverges, $q$ and the points $\omega_j$ can be chosen so that the $A_j$ together cover $C$, and likewise the $B_j$.

Dependencies

None from the paper's references; Cohen's theorem (reference [1], Amer. Math. Monthly 59 (1952), 704--705) is context only.

Bears on

• Problem 1215: the paper poses the question of that problem in connection with Theorem 1 (p. 347); the theorem itself concerns radii and says nothing about path lengths.