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.
Categories: math.CV