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Statement

For a monic polynomial $p$ of degree $d$, $E(p):=\{z:|p(z)|=1\}$ and $|E(p)|$ is its length (p. 1). The constant $\alpha_0$ is the least upper bound of the perimeters of the convex hulls of compact connected sets of logarithmic capacity $1$ (p. 1); its exact value is unknown, Pommerenke proved $\alpha_0<9.173$, and the paper reports the conjectured value $3^{3/2}2^{2/3}\approx8.24$ (p. 1).

Theorem 1 (p. 1, quoted). "For monic polynomials $p$ of degree $d$ $|E(p)|\le\alpha_0d<9.173d$."

The paper places this against the earlier bounds $|E(p)|\le74d^2$ (Pommerenke) and $|E(p)|\le8\pi ed\approx68.32d$ (P. Borwein), and against the conjectured extremal case $p(z)=z^d+1$, where $|E(p)|=2d+O(1)$ as $d\to\infty$ (p. 1). The acknowledgement (p. 8) credits the referee with improving the authors' original estimate in Theorem 1.

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 the definition of $\alpha_0$ were read clause by clause on the print, and the proof (pp. 7--8) was read in outline. Nothing here is independently reviewed.

Proof pointer

Pp. 7--8. Since the length is maximized over monic degree-$d$ polynomials (Lemma 4), it suffices to bound $|E(p)|$ for an extremal $p$, and Lemma 6 supplies one with $E(p)$ connected. As a connected set of capacity $1$, such an $E(p)$ has convex hull of perimeter at most $\alpha_0$, below $9.173$ by Pommerenke's bound (the paper's Lemma 7, p. 7). The Crofton-type integral-geometric formula writes $|E|$ as half the integral over lines of the number of intersections; a connected compact set meets exactly the lines that the boundary of its convex hull meets, almost every one of which that boundary meets twice, while $E(p)$ meets each line at most $2d$ times (Lemma 1). Comparing the two integrals gives the factor $d$.

Dependencies

• Lemma 1, Lemma 4, Lemma 6.
• Lemma 7 (p. 7), Pommerenke's bound $\pi(\sqrt{10}-3\sqrt2+4)<9.173$ for the convex-hull perimeter of a connected compact set of capacity $1$, cited from Pommerenke, Math. Ann. 139 (1959), 64--75, Satz 5.
• The integral-geometric formula for the length of a curve (Santaló).

Bears on

• #114: an upper bound only. It gives $|E(p)|<9.173\,d$ for every monic $p$ of degree $d$, against the length $2d+O(1)$ of the conjectured maximizer $z^d+1$ (equal in length to $z^d-1$, a rotation of it); it does not decide whether $z^n-1$ is the maximizer in any degree.