Geodesic
On Estimates of Lengths of Lemniscates
Matematičeskie zametki, Tome 92 (2012) no. 6, pp. 872-883

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For any natural number $n$ and any $C>0$, we obtain an integral formula for calculating the lengths $|L(P_n,C)|$ of the lemniscates $$ L(P_n,C):=\{z:|P_n(z)|=C\} $$ of algebraic polynomials $P_n(z):=z^n+c_{n-1}z^{n-1}+\dots+c_0$ in the complex variable $z$ with complex coefficients $c_j$, $j=0, \dots, n-1$, and establish the upper bound for the quantities $\lambda_n:=\sup\{|L(P_n,1)|: P_n(z)\}$, which is currently best for $3\leq n\leq10^{14}$. We also study the properties of the derivative $S'(C)$ of the area function $S(C)$ of the set $\{z:|P_n(z)|\leq C\}$.
Keywords: lemniscate of an algebraic polynomial, length of a lemniscate, conformal $n$-sheeted mapping
Mots-clés : Lebesgue measure, Jordan domain, Jordan arc. 
@article{MZM_2012_92_6_a7,
     author = {O. N. Kosukhin},
     title = {On {Estimates} of {Lengths} of {Lemniscates}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {872--883},
     publisher = {mathdoc},
     volume = {92},
     number = {6},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2012_92_6_a7/}
}
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O. N. Kosukhin. On Estimates of Lengths of Lemniscates. Matematičeskie zametki, Tome 92 (2012) no. 6, pp. 872-883. http://geodesic.mathdoc.fr/item/MZM_2012_92_6_a7/