On Estimates of Lengths of Lemniscates

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 $\{z:|P_n(z)|\leq C\}$.