Given a square Vandermonde matrix
$\mathbf{V} =$\begin{pmatrix} 1 & x_0 & x_0^2 & \ldots x_0^{n-1}\\ 1 & x_1 & x_1^2 & \ldots x_1^{n-1}\\ \vdots\\ 1 & x_{n-1} & x_{n-1}^2 & \ldots x_{n-1}^{n-1}\\ \end{pmatrix}
and an integer vector $\mathbf{a}$ of length $n$:
$\mathbf{a} = (a_0, a_1, \ldots, a_{n-1})^T$,
we can compute a vector $\mathbf{a}' = \mathbf{V}\cdot \mathbf{a}$ via matrix-vector multiplication.
In this case, what is the relationship between $\max({\mathbf{a}})$ and $\max({\mathbf{a'}})$??
What you are asking is equivalent to discussing the relation between the maximum value of a polynomial evaluated at $n-1$ points and the largest coefficient.
I don't think that there is any simple relation. For even $n$ (odd degree) or odd $n$ with a positive leading coefficient, the maximum is unbounded (asymptotic to $a_{n-1}(\max(\mathbf x))^{n-1}$).