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Danchenko 2007 lengths lemniscates
lemma_1: The total length of a multiple union of finitely many piecewise smooth curves with finite secant variation Psi(L) is at most Psi(L) times the analytic capacity of their union, with equality for a circle.
theorem_1: For every monic complex polynomial P_n of degree n and every r > 0, the lemniscate where the modulus of P_n equals r to the n has total length at most 2 pi n r.
theorem_2: On a rectifiable curve sigma with finite secant variation Psi(sigma), the integral of |R'| over the part of sigma in a compact E is at most n Psi(sigma) times the maximum of |R| there, for every rational R of degree at most n without poles on sigma.
Danchenko, V. I., The lengths of lemniscates. Variations of rational functions. Mat. Sb. 198 (2007), no. 8, 51--58. The file prints "© В. И. Данченко, 2007", the author's copyright line, at the foot of its first page (read from the text layer, which renders the symbol as "c⃝") and no license wording on any of its eight pages, and the hosting site's terms of use state that its materials "are fully copyrighted by Steklov Mathematical Institute, Russian Academy of Sciences, and/or by other copyright holder" and that reproduction or republication "requires written permission of the copyright holder" (https://www.mathnet.ru/php/agreement.phtml?option_lang=eng, read 2026-10-02), every other right reserved.
Written in Russian, the paper studies the extremal problem of Erdos, Herzog and Piranian (1958) on the length of a lemniscate L(P_n,r) = {z : |P_n(z)| = r^n} for a monic degree-n polynomial. Theorem 1 proves |L(P_n,r)| <= 2 pi n r, halving Dolzhenko's bound 4 pi n r and improving Eremenko-Hayman's 9.173n and Borwein's 8 pi e n at r = 1. The proof is short: Lemma 1 bounds the length of a multiple union of piecewise smooth curves by Psi(L) gamma(L), the secant variation times the analytic capacity, and Lemmas 2-3 estimate these two quantities for lemniscates. Theorem 2 is a companion sharp estimate for rational functions: for a rectifiable curve sigma with finite secant variation Psi(sigma) and any rational R of degree at most n with no poles on sigma, the L^1 norm of R' over the part of sigma in a compact E is at most n Psi(sigma) times the maximum of |R| there; for sigma inside E, var_sigma R <= n Psi(sigma) ||R||_C <= 2n(pi + Phi(sigma)) ||R||_C, the last bound meaningful for Radon curves; the first of these is an equality for R(z)=z^n on a circle. The author records the natural conjecture that the Bernoulli-type lemniscate |z^n - 1| = 1, of length 2n + 4 log 2 + O(1/n), is the longest of the L(P_n,1), which is the question of Erdos problem 114.
Source: https://www.mathnet.ru/eng/sm3795.
Read status. Claims checked: Theorem 1 (p. 52), Lemma 1 (p. 53) with Remark 1 and Lemma 1a (pp. 55-56), Theorem 2 (pp. 56-57) and the Bernoulli length computation (p. 52) were read clause by clause on the print. The proofs were read but not checked step by step.
Bears on. #114: Theorem 1 at r = 1 bounds the length of {|p(z)| = 1} by 2 pi n for every monic p of degree n, against the length 2n + 4 log 2 + O(1/n) of the conjectured maximiser z^n - 1; it does not decide whether z^n - 1 maximises the length.
Results. Theorem 1 (p. 52), with the Bernoulli lemniscate's length; Lemma 1 (p. 53), with Remark 1 and Lemma 1a (pp. 55-56); Theorem 2 (pp. 56-57). Lemmas 2 and 3 (p. 55) and Lemma 4 (p. 56) are proof steps, summarized on the pages of the theorems they serve.
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