For $d, m \in\mathbb{N}$ fixed, let $P\equiv P(x) := \sum_{|\alpha|\leq m} c_\alpha\cdot x^\alpha$ be a real polynomial in $d$ variables of (total) degree $m$. (That is, the above sum ranges over all multiindices $\alpha=(i_1, \ldots, i_d)\in\mathbb{N}_0^{\times d}$ of length $|\alpha|\equiv i_1+\ldots + i_d$ less than $m$.)

Is it possible to estimate the maximum coefficient $\|c_\alpha\|_\infty := \max_{|\alpha|\leq m}|c_\alpha|$ of $P$ against its uniform norm $\|P\|_{\infty;K}:= \sup_{x\in K}|P(x)|$, for $K$ some compact set in $\mathbb{R}^d$?

That is, does there exist a compact set $K$ in $\mathbb{R}^d$ together with a constant $\kappa \equiv \kappa(m,d,K)>0$ such that

$$\tag{1}\|c_\alpha\|_\infty \ \leq \ \kappa\cdot \|P\|_{\infty; K} \qquad \text{ for each } \ P \ \text{ as above}?$$

Any references are welcome.

**Edit:** Do you know if $\kappa$ can be chosen independent of $m$, or at least such that the sequence $(\kappa(m,d,K))_{m\geq 0}$ is bounded (for $K$ and $d$ fixed)?

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