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 $$L\left( {P_n ,C} \right): = \left\{ {z:\left| {P_n \left( z \right)} \right| = C} \right\}$$ of algebraic polynomials P _n (z):= z ⁿ + c _n?1 z ^n?1 + ... + c ₀ in the complex variable z with complex coefficients c _j , j = 0, ..., n ? 1, and establish the upper bound for the quantities $$\lambda _n : = \sup \left\{ {\left| {L\left( {P_n ,1} \right)} \right|:P_n (z)} \right\},$$ which is currently best for 3 ≤ n ≤ 10¹⁴. We also study the properties of the derivative S′(C) of the area function S(C) of the set {z: |P _n (z)| ≤ C}.