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Statement

Extremal means maximizing $|E(p)|$ among all monic polynomials of degree $d$, as on the Lemma 5 page.

Lemma 6 (p. 7, quoted). "There exists an extremal polynomial $p$ for which the set $E(p)$ is connected."

Remark after Lemma 6 (p. 7). Moving the critical values of modulus greater than $1$ towards infinity instead of to the unit circle, and using the arguments of the proof of Lemma 4, one can show that an extremal polynomial has no critical value of modulus greater than $1$; hence $E(p)$ is connected for every extremal $p$. The remark is a sketch, and the paper says it does not use it in the proof of Theorem 1.

The remarks on p. 2 record that P. Borwein had observed that his method would give $|E(p)|\le4\pi d\approx12.57d$ if one knew that $E(p)$ is connected for extremal $p$; those remarks cite this fact as "our Lemma 3" [sic], the lemma printed as Lemma 6.

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, the remark and the proof (p. 7) were read on the print. Nothing here is independently reviewed.

Proof pointer

P. 7. Take the extremal $p$ of Lemma 5, all of whose critical values lie on the unit circle. Then $p$ maps $D=\{z\in\overline{\mathbb C}:|p(z)|>1\}$ onto the exterior of the unit disc as a branched covering of degree $d$ whose only critical point is $\infty$, of index $d-1$; the Riemann–Hurwitz formula makes $D$ simply connected, so its boundary $E(p)$ is connected.

Dependencies

Lemma 5; the Riemann–Hurwitz formula.

Bears on

• #114: a structural reduction. It lets the maximal length in each degree be sought among monic polynomials with connected lemniscate, as that of $z^n-1$ is; it decides the question in no degree on its own. The pending degree-3 claim Chatelet 2026 cites a reduction attributed in part to this paper; that claim page records how its use compares with what Lemmas 5 and 6 give.