---

Statement

Lemma 1 (p. 2, quoted). "For every rational function $f$ of degree $d$ the $f$-preimage of any line or circle has no more than $2d$ intersections with any line or circle $C$, except finitely many $C$'s."

The paper calls this the main property of the level sets $E(p)$ (p. 2): for a polynomial $p$ of degree $d$, $E(p)$ is the $p$-preimage of the unit circle. The proof shows that the only exceptional $C$ are those contained in the preimage; as $E(p)$ is bounded it contains no line, so $E(p)$ meets each line at most $2d$ times, the form used in the proof of Theorem 1 (p. 8).

Source. Alexandre Eremenko and Walter Hayman, On the length of lemniscates, Michigan Math. J. 46 (1999), no. 2, 409--415, DOI 10.1307/mmj/1030132418; page numbers are those of the authors' corrected preprint (pp. 1--9) named on the source card, not the journal's pagination.

Read depth. Claims checked: the statement and its proof (p. 2) were read on the print. Nothing here is independently reviewed.

Proof pointer

P. 2. Fractional-linear maps act transitively on circles of the Riemann sphere and preserve the degree under composition, so one may take both circles to be the real line. A real point $z_0$ with $f(z_0)$ real is a zero of $f(z)-\overline{f(\bar z)}$, a rational function of degree at most $2d$, which has at most $2d$ zeros unless it vanishes identically, that is, unless the whole line lies in the preimage.

Dependencies

None in the paper.

Bears on

• #114: an ingredient of the upper bound of Theorem 1 (at most $2d$ crossings of a line by $E(p)$) and of Theorem 2; it says nothing about which polynomial maximizes the length.