On the size of lemniscates of polynomials in one and several variables

In the convergence theory of rational interpolation and Padé approximation, it is essential to estimate the size of the lemniscatic set $E:=\big \{z\,:\, |z|\le r$ and $|P(z)|\le \epsilon ^{n}\big \}$, for a polynomial $P$ of degree $\le n$. Usually, $P$ is taken to be monic, and either Cartan's Lemma or potential theory is used to estimate the size of $E$, in terms of Hausdorff contents, planar Lebesgue measure $m_{2}$, or logarithmic capacity cap. Here we normalize $\|P\|_{L_{\infty }\bigl (|z|\le r\bigr )}=1$ and show that cap$(E)\le 2r\epsilon$ and $m_{2} (E)\le \pi (2r\epsilon )^{2}$ are the sharp estimates for the size of $E$. Our main result, however, involves generalizations of this to polynomials in several variables, as measured by Lebesgue measure on $\mathbb {C}^{n}$ or product capacity and Favarov's capacity. Several of our estimates are sharp with respect to order in $r$ and $\epsilon$.