Bernstein–Walsh inequalities and the exponential curve in $\mathbb {C}^2$

Abstract:

It is shown that for the pluripolar set $K=\{(z,e^z): |z|\leq 1\}$ in ${\Bbb C}^2$ there is a global Bernstein–Walsh inequality: If $P$ is a polynomial of degree $n$ on ${\Bbb C}^2$ and $|P|\leq 1$ on $K$, this inequality gives an upper bound for $|P(z,w)|$ which grows like $\exp (\frac 12n^2\log n)$. The result is used to obtain sharp estimates for $|P(z,e^z)|$.