  Let ${{\bf x}}=(x_1,\dots,x_d) \in [-1,1]^d$ be linearly independent over
$\mathbb Z$, set $K=\{(e^{z},e^{x_1 z},e^{x_2 z}\dots,e^{x_d z}): |z| \le 1\}.$
We prove sharp estimates for the growth of a polynomial of degree $n$, in terms
of $$E_n({\bf x}):=\sup\{\|P\|_{\Delta^{d+1}}:P \in \mathcal P_n(d+1), \|P\|_K
\le 1\},$$ where $\Delta^{d+1}$ is the unit polydisk. For all ${{\bf x}} \in
[-1,1]^d$ with linearly independent entries, we have the lower estimate $$\log
E_n({\bf x})\ge \frac{n^{d+1}}{(d-1)!(d+1)} \log n - O(n^{d+1});$$ for
Diophantine $\bf x$, we have $$\log E_n({\bf x})\le \frac{
n^{d+1}}{(d-1)!(d+1)}\log n+O( n^{d+1}).$$ In particular, this estimate holds
for almost all $\bf x$ with respect to Lebesgue measure.
