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Statement

Lemma 4 (p. 3, quoted). "The length $|E(p)|$ is a continuous function of the coefficients of $p$. For every positive integer $d$ there exists a monic polynomial $p_d$ with the property $|E(p_d)|\ge|E(p)|$ for every monic polynomial $p$ of degree $d$."

Here $E(p)=\{z:|p(z)|=1\}$ (p. 1). After the proof the paper calls extremal any polynomial that maximizes $|E(p)|$ among all monic polynomials of degree $d$ (p. 5); this lemma says extremal polynomials exist.

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 was read on the print and the proof (pp. 3--5) was read in outline. Nothing here is independently reviewed.

Proof pointer

Pp. 3--5. Writing $p=p_Z$ by its zero vector $Z\in\mathbb C^d$, the proof first shows $|E(p_Z)|\to0$ as the diameter of $Z$ tends to infinity: for large diameter the zeros split into two far-apart clusters, Cartan's lemma (Lemma 3, p. 3) puts $E(p)$ inside discs of small total radius, and the Corollary after Lemma 2 (p. 3) turns small projections into small length. Continuity in $Z$ follows by writing $|E(p)|$ as an integral over the unit circle of the summed moduli of the derivatives of the branches of $p^{-1}$, with a uniform-integrability estimate near the at most $d-1$ critical values on the circle (again from Cartan's lemma). A continuous function tending to $0$ at infinity attains its maximum, and the zeros depend continuously on the coefficients.

Dependencies

• Lemma 2 and its Corollary (pp. 2--3): an analytic curve crossing each horizontal and vertical line at most $n$ times has length at most $n$ times the sum of its two projections, and a connected subset $l$ of $E(p)$ satisfies $|l|\le4d\operatorname{diam}(l)$.
• Lemma 3 (Cartan's lemma, p. 3, cited from Levin): for a monic $p$ of degree $d$, $\{z:|p(z)|<M\}$ lies in a union of discs whose radii sum to $2eM^{1/d}$.
• Continuity of algebraic functions (Hille, Theorem 12.2.1).

Bears on

• #114: background only. It shows the maximum the problem asks about is attained in every degree; it does not identify the maximizer.