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

Abstract:

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 $\epsilon$.