Eremenko 1999 length lemniscates

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lemma_1: For every rational function f of degree d, the f-preimage of any line or circle meets every line or circle C in at most 2d points, except for finitely many C.

lemma_4: The length of E(p) is a continuous function of the coefficients of p, and for every positive integer d some monic polynomial of degree d has a level set at least as long as that of every monic polynomial of degree d.

lemma_5: For each degree d, some monic polynomial that maximizes the length of E(p) among all monic polynomials of degree d has all its critical points in E(p).

lemma_6: For each degree d, some monic polynomial that maximizes the length of E(p) among all monic polynomials of degree d has E(p) connected; a remark sketches that every extremal polynomial has this property.

remark_p5: The remark after Lemma 5 deduces that z^2 + 1 maximizes the length of the level set where a monic quadratic has modulus one; that level set is the Bernoulli lemniscate, of length about 7.416.

theorem_1: For every monic polynomial p of degree d the level set where p has modulus one has length at most alpha_0 d, which is less than 9.173 d, where alpha_0 is the supremum of the convex-hull perimeters of continua of capacity one.

theorem_2: For a rational function f of degree d, the spherical length of the f-preimage of any circle is at most d times the length of a great circle, with equality for f(z) = z^d and C the real line.

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Eremenko, Alexandre and Hayman, Walter, On the length of lemniscates. Michigan Math. J. 46(2) (1999), 409--415. DOI 10.1307/mmj/1030132418. The copy read for this card is the authors' corrected preprint from the first author's papers page, which prints no copyright, license or terms line on any of its nine pages; that page states no copyright, license or terms either (https://www.math.purdue.edu/~eremenko/papers.html), and the journal edition's terms do not govern that manuscript; the term is unstated.

For a monic polynomial p of degree d the paper studies E(p) = {z : |p(z)| = 1} and proves in Theorem 1 that its length satisfies |E(p)| <= alpha_0 d < 9.173 d, where alpha_0 is the supremum, over compact connected sets K of logarithmic capacity 1, of the perimeter of the convex hull of K (Pommerenke proved alpha_0 < 9.173); this improves Pommerenke's 74 d^2 and Borwein's 8 pi e d ~ 68.32 d. For d = 2 the extremal level set is shown to be the Bernoulli lemniscate. A key ingredient is Lemma 6 (Borwein had noted that his method would give 4 pi d if this were known): some extremal polynomial has a connected level set E(p) (a remark sketches why every extremal one does); the proof also uses Lemma 1 (for a rational f of degree d, the f-preimage of any line or circle meets any line or circle C in at most 2d points, except for finitely many C), Lemma 2 (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 Cartan's lemma (Lemma 3). Theorem 2 solves the analogous rational problem completely: the spherical length of the f-preimage of any circle is at most d times the length of a great circle, which is sharp for f(z) = z^d and C the real line. The paper directly addresses the Erdős-Herzog-Piranian conjecture that |E(p)| is maximal for p(z) = z^d + 1, the subject of problem 114, and settles it only for d = 2.

Source: https://www.math.purdue.edu/~eremenko/papers.html.

Bears on. #114: the remark after Lemma 5 (p. 5) settles degree 2, $z^2+1$ (a rotation of $z^2-1$) being extremal among monic quadratics; Theorem 1 is an upper bound $|E(p)|<9.173\,d$ in every degree; Lemmas 5 and 6 show that in each degree some maximizer has all critical values on the unit circle and a connected level set. No degree other than 2 is decided.

Results. Page numbers are those of the preprint read (pp. 1--9).

• Theorem 1 (p. 1): for monic $p$ of degree $d$, $|E(p)|\le\alpha_0d<9.173d$.
• Theorem 2 (p. 2): for a rational $f$ of degree $d$, the spherical length of the preimage of any circle is at most $d$ great-circle lengths; sharp for $z^d$ and the real line.
• Lemma 1 (p. 2): the preimage of a line or circle under a degree-$d$ rational map meets any line or circle in at most $2d$ points, apart from finitely many exceptions.
• Lemma 4 (p. 3): $|E(p)|$ is continuous in the coefficients, and a maximizer exists in each degree.
• Lemma 5 (p. 5): some extremal polynomial has all its critical points in $E(p)$.
• Remark after Lemma 5 (p. 5): $z^2+1$ is extremal for $d=2$, its level set the Bernoulli lemniscate of length about $7.416$.
• Lemma 6 (p. 7): some extremal polynomial has a connected level set $E(p)$; a remark after it sketches that $E(p)$ is connected for every extremal $p$.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.