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)?



Inequality between coefficients of a polynomial and its supremum
Yes, such a $\kappa$ exists for every compact set $K$ with non-empty interior. Here is an abstract linear-algebra argument.


Let $V$ be the real vector space spanned by the multi-indices $\alpha$ with length at most $m$.
We have a linear map $A\colon V \to \mathbb R^K$, sending $(c\_\alpha)$ to the function $x\to \sum\_\alpha c\_\alpha x^\alpha$. Since different polynomials cannot agree on a set with non-empty interior, this map is injective. There is then a finite subset $F\subset K$ such that the composition $V\xrightarrow A\mathbb R^K\to\mathbb R^F$ is a linear isomorphism (for example, pick the elements of $F$ one by one, making sure that the rank of the composition increases by 1 each time).


Now $\kappa$ exists by the fact that all linear isomorphisms between finite-dimensional vector spaces are bicontinuous.


If you want an explicit value for $\kappa$, you can use discrete derivatives (see <https://en.wikipedia.org/wiki/Finite_difference>) to write $c\_\alpha$ as a linear combination of some values of $P$.


