Low rank approximation of polynomials

Abstract

Let $k\leq n$. Each polynomial $p\in\oR[x_1,...,x_n]$ can be uniquely written as $p=\sum_{\mu}\mu p_{\mu}$, where $\mu$ ranges over the set $M$ of all monomials in $\oR[x_1,...,x_k]$ and where $p_{\mu}\in\oR[x_{k+1},...,x_n]$. If $p$ is $d$-homogeneous and $\varepsilon>0$, we say that $p$ is {\em $\varepsilon$-concentrated on the first $k$ variables} if $$\sum_{\mu\in M\atop\deg(\mu)<d}\max_{x\in\oR^{n-k}\atop\|x\|=1}p_{\mu}(x)^2\leq\varepsilon\|p\|^2,$$ where $\|p\|$ is the Bombieri norm of $p$. We show that for each $d\in\oN$ and $\varepsilon>0$ there exists $k_{d,\varepsilon}$ such that for each $n$ and each $d$-homogeneous $p\in\oR[x_1,...,x_n]$ there exists $k\leq k_{d,\varepsilon}$ such that $p$ is $\varepsilon$-concentrated on the first $k$ variables {\em after some orthogonal transformation of $\oR^n$}. (So $k_{d,\varepsilon}$ is independent of the number $n$ of variables.) We derive this as a consequence of a more general theorem on low rank approximation of polynomials.