I was looking in the literature but I couldn't find a nice expression; is there a simple expression for the $C^k$-norm of a polynomial of degree $d>k$?
Here by the $C^k$ norm, I mean the maximum formulation: $$ \|\|f\|\| := \max_{|\beta|\le k}\,\sup_{x\in \Omega}\, \|D^{\beta}f(x)\| $$ for a sufficiently smooth function where $\Omega$ is a simple connected domain n $\mathbb{R}^d$ with Lipschitz boundary.