Estimate the coefficients of a polynomial against its maximum
Solution 1:
One thing you can do is pick out the coefficient $c_\alpha$ for any $\alpha$ using some variant of the Cauchy integral formula. By $d$ applications of Cauchy we have $$\frac{1}{(2\pi i)^d} \int_{(S^1)^d} P(z) z^{-\alpha-1} dz = c_\alpha.$$ It follows that $$|c_\alpha| \leq \max_{z \in (S^1)^d} |P(z)|.$$
I doubt you can get away with a compact subset of $\mathbb{R}^d$, even if $d=1$.