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Statement

Setting (p. 3). $p$ is a monic polynomial of degree $n\ge2$ (the paper sets aside $n=1$ as trivial), $L=L_p=\{\lvert p(z)\rvert=1\}$ and $E=E_p=\{\lvert p(z)\rvert<1\}$.

Improved upper bound (Section 8, pp. 10--11; the bound on p. 11). For every such $p$,

$$\lvert L\rvert\le2\pi(n-1+\sqrt n).$$

The paper gives no theorem number and calls the bound "only marginally worse than Danchenko's estimate $2\pi n$" (p. 11).

The simplest upper bound (Section 7, p. 10). With the first extension of the normal, the same method gives $\lvert L\rvert\le2\pi(2n-1)$, and $\lvert L\rvert\le2\pi(2k-1)$ when $p$ has $k$ distinct roots (p. 10).

The length formula (p. 11, (6)). With $\varphi=p'/p$ and $\psi=p''/p'=\sum_\zeta(z-\zeta)^{-1}$, the sum over the roots $\zeta$ of $p'$ counted with multiplicity,

$$\lvert L\rvert=2\iint_E\lvert p'\rvert\,dA -\iint_E\frac{\lvert p\varphi\rvert}{\varphi}\,\psi\,dA .$$

Proof pointer

P. 11. Since $p$ covers the unit disk $n$ times on $E$, $\iint_E\lvert p'\rvert^2\,dA=\pi n$, and Cauchy's inequality with $A(E)\le\pi$ bounds the first term of (6) by $2\pi\sqrt n$. The second term is at most $\sum_\zeta\iint_E\lvert z-\zeta\rvert^{-1}\,dA$, and each of the $n-1$ summands is at most $\iint_{\mathbb D}\lvert z\rvert^{-1}\,dA=2\pi$ because $E$ has logarithmic capacity $1$ and hence area at most $\pi$ (Pólya's theorem, used on p. 10). Formula (6) itself comes from Stokes' formula, $\lvert L\rvert=2\operatorname{Re}\iint_E\partial s\,dA$ (p. 4, (4)), for the extension $s=\lvert p'\rvert/\varphi$ of the outward unit normal (p. 10).

Read depth

Claims checked: the statements of Sections 7 and 8 and formula (6) were read clause by clause on the page images of the arXiv version, and the proof on p. 11 was followed. Nothing here is independently reviewed.

Dependencies

None in the corpus. External input named by the paper: Pólya's area bound for sets of logarithmic capacity (Ransford, Theorem 5.3.5).

Source. A. Fryntov and F. Nazarov, New estimates for the length of the Erdős-Herzog-Piranian lemniscate, in Linear and Complex Analysis, Amer. Math. Soc. Transl. Ser. 2, 226 (2009), 49--60, doi:10.1090/trans2/226/05; the edition read, and its page numbering, are named on the source card.

Bears on

• Problem 114: an upper bound $2\pi(n-1+\sqrt n)$ for the lemniscate length of every monic polynomial of degree $n\ge2$, weaker than Danchenko's $2\pi n$ for $n\ge2$; it decides the question for no degree.