# Estimate the homogeneous components of a polynomial against its maximum

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.

• It looks like in several places you write $m$ where you mean to write $k$. As written the definition of $P_k$ does not depend on $k$. It would also be nice if instead of $||(P_m)||_{\infty;K}$ you just wrote $||(P)||_{\infty;k}$, since the former makes it look like it just depends on the $m$-th homogeneous component. Commented Dec 18, 2021 at 19:54
• @Vik78 You're right, thanks for pointing that out. I edited accordingly. As for your comment re $(P_k)$, I wrote it that way to indicate that the norms in question refer to (the norms of the components of) the degree-based gradation $(P_k)$ of $P$. Commented Dec 18, 2021 at 19:56
• Now you have changed its definition to read $||(P_k)||_{\infty;K}$ with $k$ as a subscript— but you are maximizing over $0 \le k \le m$ so it does not depend on $k$. Personally I find it confusing (and would just replace the entire symbol $||(\cdot)||$ with a function $f(P, K)$, since as written it looks very similar to the norm $||\cdot||$). Commented Dec 18, 2021 at 20:04
• Do you want to have the constant $\kappa$ independent of the degree of the polynomial? This looks very unlikely. Commented Dec 18, 2021 at 21:10
• @JochenWengenroth Yes, $\kappa$ should be independent of the degrees of $P$ and $P_k$. Commented Dec 18, 2021 at 21:12

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.

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