Let $P\equiv P(x) := \sum\_{|\alpha|\leq m} c\_\alpha\cdot x^\alpha$ be a real polynomial in $d$ variables of (total) degree $m$, where $d, m \in\mathbb{N}$ are fixed.


(I.e., 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 $m$.)


Denote by $P\_k = \sum\_{|\alpha|=m}c\_\alpha\cdot x^\alpha$, $ 0\leq k\leq m$, the $k$-th homogeneous component of $P$.


I was wondering the following: Given a compact subset $K$ in $\mathbb{R}^d$, is it possible to for $\varphi\_K(P):=\max\_{0\leq k\leq m}\|P\_k\|\_{\infty; K}$ (or indeed for any $\ell\_p$-norm of $(\|P\_k\|\_{\infty;K})\_{k\geq 0}$ with $1\leq p \leq \infty$) find a constant $\kappa=\kappa(d,K)$ such that


$$\tag{1} \varphi\_K(P) \ \leq \ \kappa\cdot \|P\|\_{\infty; K} \qquad \text{ for each } \ P \ \text{ as above} \ ?$$


(Here, $\|f\|\_{\infty;K}:=\sup\_{x\in K}|f(x)|$ is the uniform norm over $K$.) Any references are welcome.



Estimate the homogeneous components of a polynomial against its maximum
The answer is no. E.g., let $d=1$, $K=[0,1]$, and, for $x\in K$,
$$P(x):=T\_n(x):=n\sum\_{0\le k\le n/2}\frac{(-1)^k}{n-k}\binom{n-k}k2^{n-2k-1}x^{n-2k}
=\cos(n\arccos x),$$
the $n$th [Chebyshev polynomial](https://en.wikipedia.org/wiki/Chebyshev_polynomials#Explicit_expressions).


Then $\|P\|\_{\infty;K}\le1$, whereas (say) for $k=0$ we have $\|P\_k\|\_{\infty;K}=2^{n-1}\to\infty$ as $n\to\infty$.


