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Statement

A monic polynomial of degree $d$ is extremal when it maximizes $|E(p)|$, the length of $E(p)=\{z:|p(z)|=1\}$, among all monic polynomials of degree $d$ (p. 5); extremal polynomials exist by Lemma 4.

Lemma 5 (p. 5, quoted). "There exists an extremal polynomial $p$, such that all critical points of $p$ are contained in $E(p)$."

Equivalently, all critical values of that $p$ lie on the unit circle (p. 7). The lemma asserts existence of one such extremal polynomial, not that every extremal polynomial has the property; the remark after Lemma 6 (p. 7) sketches a fact about every extremal polynomial, that none has a critical value of modulus greater than $1$. Its consequence for $d=2$ is recorded on the remark on p. 5.

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. 5--7) was read in outline. Nothing here is independently reviewed.

Proof pointer

Pp. 5--7. A critical value $a$ off the unit circle can be moved to $a+\lambda$, keeping the other critical values fixed, through a family of monic polynomials $p_\lambda$ obtained from a quasiconformal deformation supported in a small disc about $a$ and the measurable Riemann mapping theorem with analytic dependence on parameters. For extremal $p$ and a disc missing the unit circle, $|E(p_\lambda)|$ is the integral over $E(p)$ of the modulus of an analytic function of $\lambda$, hence subharmonic in $\lambda$; it is maximal at $\lambda=0$ and so constant. Moving each critical value off the circle along disjoint curves to the circle therefore keeps the length and ends at an extremal $p^*$ with all critical values on the circle.

Dependencies

• Lemma 4 (continuity of $|E(p_\lambda)|$ in $\lambda$).
• The existence and analytic-dependence theorems for the Beltrami equation (cited from Carleson and Gamelin, Ch. I, Theorems 7.4 and 7.6).

Bears on

• #114: a structural reduction. It restricts the search for one maximizer in each degree to polynomials whose critical values all lie on the unit circle, as $z^n-1$ does; for $d=2$ it decides the question (the remark on p. 5), and for $d\ge3$ it does not.