Non-symmetric polarization

TL;DR

This paper investigates the relationship between the norms of non-symmetric multilinear forms derived from homogeneous polynomials and the original polynomials, establishing bounds that depend logarithmically on the number of variables.

Contribution

It provides new estimates for the norms of non-symmetric multilinear forms associated with homogeneous polynomials, extending understanding of their behavior under various norms.

Findings

Norms of non-symmetric multilinear forms can be estimated by polynomial norms with a logarithmic factor.

For -norms with p<2, the logarithmic factor is unnecessary.

The results apply to a broad class of norms on complex vector spaces.

Abstract

Let $P$ be an $m$-homogeneous polynomial in $n$-complex variables $x_1, \dotsc, x_n$. Clearly, $P$ has a unique representation in the form \begin{equation*} P(x)= \sum_{1 \leq j_1 \leq \dotsc \leq j_m \leq n} c_{(j_1, \dotsc, j_m)} \, x_{j_1} \dotsb x_{j_m} \,, \end{equation*} and the $m$"~form \begin{equation*} L_P(x^{(1)}, \dotsc, x^{(m)})= \sum_{1 \leq j_1 \leq \dotsc \leq j_m \leq n} c_{(j_1, \dotsc, j_m)} \, x^{(1)}_{j_1} \dotsb x^{(m)}_{j_m} \end{equation*} satisfies $L_P(x,\dotsc, x) = P(x)$ for every $x\in\mathbb{C}^n$. We show that, although $L_P$ in general is non-symmetric, for a large class of reasonable norms $ \lVert \cdot \rVert $ on $\mathbb{C}^n$ the norm of $L_P$ on $(\mathbb{C}^n, \lVert \cdot \rVert )^m$ up to a logarithmic term $(c \log n)^{m^2}$ can be estimated by the norm of $P$ on $ (\mathbb{C}^n, \lVert \cdot \rVert )$; here $c \ge 1$ denotes a universal constant. Moreover, for the $\ell_p$"~norms $ \lVert \cdot \rVert_p$, $1 \leq p < 2$ the logarithmic term in the number $n$ of variables is even superfluous.