Fryntov 2009 new estimates length erdos herzog

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estimate_p11: Fryntov and Nazarov's bound for every degree: the lemniscate |p(z)|=1 of a monic polynomial p of degree n >= 2 has length at most 2 pi (n - 1 + sqrt n), improving the bound 2 pi (2n - 1) of their Section 7.

estimate_p17: Fryntov and Nazarov's asymptotic bound: the lemniscate |p(z)|=1 of every monic polynomial p of degree n has length at most 2n + O(n^{7/8}) as n tends to infinity, which the abstract states as 2n + o(n).

local_maximum_p4: Fryntov and Nazarov's local result: |L_p| <= |L_{p_0}| for every monic polynomial p of degree n sufficiently close to p_0(z) = z^n - 1, so the lemniscate length attains a local maximum at p_0.

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Fryntov, Alexander and Nazarov, Fedor, New estimates for the length of the {E}rdős-{H}erzog-{P}iranian lemniscate. In Linear and Complex Analysis, Amer. Math. Soc. Transl. Ser. 2, 226 (2009), 49--60. DOI 10.1090/trans2/226/05. The copy read for this card is the arXiv preprint arXiv:0808.0717v1 (dated May 8, 2008), and the page numbers below are that copy's. The arXiv record names arXiv's non-exclusive distribution license, every other right reserved.

Erdős, Herzog and Piranian asked in 1958 (their Problem 12) whether, among monic polynomials $p$ of degree $n$, the lemniscate $L_p=\{z:\lvert p(z)\rvert=1\}$ is longest for $p_0(z)=z^n-1$, whose length is $2n+O(1)$ (p. 1, (1)). The authors write the length as an area integral over $E_p=\{\lvert p\rvert<1\}$ by Stokes' formula applied to an extension of the outward unit normal of $L_p$ (p. 4, (4)), and use it for two new results: $\lvert L_p\rvert\le\lvert L_{p_0}\rvert$ whenever $p$ is sufficiently close to $p_0$, so that $p_0$ is a local maximum (stated p. 4, proved in Section 6, pp. 7--10), and $\lvert L_p\rvert\le2n+O(n^{7/8})$ for every monic $p$ of degree $n$ (Section 9, pp. 11--17), which the abstract announces as $2n+o(n)$. On the way the same formula gives the bounds $2\pi(2n-1)$ (Section 7, p. 10) and $2\pi(n-1+\sqrt n)$ (Section 8, p. 11) for every $n\ge2$. The local result rests on Lemma 1 (p. 5), a length bound $2nr-c_n$ for the curve $\{\operatorname{Re}p=0\}$ in the disk of radius $r\ge2$, for $p(z)=z^n+a_2z^{n-2}+\cdots+a_n$ with $a_n$ real and $\max_{2\le k\le n}\lvert a_k\rvert=1$, and the asymptotic one on Lemma 2 (p. 14), an oscillatory-integral bound over squares for a harmonic phase.

The introduction (pp. 1--2) traces the earlier upper bounds: Dolzhenko's $4\pi n$ (thesis 1960, published 1963), Pommerenke's $74n^2$ (1961), Borwein's $8e\pi n$ (1995), Eremenko and Hayman's $9.173n$ (1999), with the case $n=2$ and, as the paper reports it, a proof that all critical points of the extremal polynomial lie on the lemniscate, and Danchenko's $2\pi n$ (2007). The full conjecture is left open, and the authors expect the exponent $7/8$ to be improvable but call going below $1/2$ "quite a challenging problem" (p. 17).

Source: https://arxiv.org/abs/0808.0717.

Read status: claims checked for the three results below, the Section 7 bound, formula (6) and Lemma 1 with its rescaled form, read clause by clause on the page images of the arXiv version; the proof of the Section 8 bound followed, those of the local maximality and the asymptotic estimate read for structure. Nothing here is independently reviewed. Result pages: local_maximum_p4, estimate_p11 and estimate_p17.

Bears on. #114: the paper proves that $z^n-1$ is a local maximizer of the lemniscate length among monic polynomials of degree $n$ (local maximality, in a neighbourhood it does not quantify) and that every such lemniscate has length at most $2n+O(n^{7/8})$ (asymptotic estimate), which matches the conjectured maximum $2n+O(1)$ to first order. Neither result decides the question for any degree.

Results.

• Local maximality (p. 4, proved in Section 6, pp. 7--10): $\lvert L_p\rvert\le\lvert L_{p_0}\rvert$ for every monic $p$ of degree $n$ sufficiently close to $p_0(z)=z^n-1$.
• Improved upper bound (Section 8, pp. 10--11): $\lvert L_p\rvert\le2\pi(n-1+\sqrt n)$ for every monic $p$ of degree $n\ge2$, with the Section 7 bound $2\pi(2n-1)$ and the length formula (6).
• Asymptotic estimate (Section 9, pp. 11--17): $\lvert L_p\rvert\le2n+O(n^{7/8})$ for every monic $p$ of degree $n$.

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