TITLE: Bounds on coefficients of factors of a multivariate polynomial
QUESTION [8 upvotes]: Given a multivariate polynomial $F(x, y, ..)$ what is the smallest bound B that can be quickly found such that $|G|_{\infty} \le B$ for all factors $G$ of $F$. (I'm using $|G|_{\infty}$ to denote the largest coefficient in G.)
I am only aware of the Landau-Mignotte bound for polynomials in $\mathbb{Z}[x]$.

REPLY [6 votes]: Mahler invented what is now known as Mahler measure to establish inequalities like this. His paper "On some inequalities for polynomials in several variables" (J. London Math. Soc. 37, 341-344, 1962) shows that if $f$ and $g$ are polynomials in $\mathbb{C}[X_1,\ldots,X_n]$ whose degrees in $X_i$ are bounded by $m_i$, then
$$
   |f|_\infty|g|_\infty\le2^{m_1+\cdots+m_n}\{(m_1+1)\cdots(m_n+1)\}^{1/2}|fg|_\infty .
$$
The arguments are elementary, and build via induction on the single variable case Mahler established in an earlier paper. The basic point is that Mahler measure is multiplicative, and can be related to height and length.
His main interest in this was its use in transcendence theory, and in particular some inequalities used by Gelfond.