I have to provide a (Markov or Bernstein-based?) inequality that gives an upper bound for the partial derivatives of a multivariate polynomial calculated near a real smooth surface in terms of value of the polynomial calculated on the surface.

More precisely: suppose $p$ a polynomial of $d$ real variables of fixed total degree $n$, $M$ a compact surface in $\mathbb{R}^{d}$ (as smooth and good as we need), i'm lookin for something near to $${\| \partial_i p\|}_{M+B(0,\epsilon)} \leq \text{Something}(n,d, \epsilon) \|p \|_M $$ where $M+B(0,\epsilon)$ means an $\epsilon$-neighborhood of $M$ and $\|f\|_K$ is the supnorm of $f$ on $K$.

There is a tangential version of Markov inequality... seems to me to not suffice...

Thank you for any suggestions!

estimate, notextimate, in the title. Perhaps someone with editing clout could fix it? – drbobmeister Jan 11 '11 at 0:57