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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 $\mathbb{Z}[x]$.

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2 Answers 2

There's a bound in Lang's book Fundamentals of Diophantine Geometry that's not great in terms of its dependence on the degree and number of variables, but it's very explicit. (Lang works with any field containing $\mathbb{Q}$, but I'll just state it for $\mathbb{C}$ coefficients).

Proposition 2.3 (page 57) Let $d\ge0$. Let $f$ and $g$ be polynomials in $\mathbb{C}[X_1,\ldots,X_n]$ satisfying $\deg(f)+\deg(g)<d$. Then $$ \frac{1}{4^{d^n}}|fg|_\infty \le |f|_\infty|g|_\infty \le 4^{d^n}|fg|_\infty. $$

If one instead uses a $p$-adic absolute value, the same estimate is true without the $4^{d^n}$ factors, i.e., $|fg|_p=|f|_p|g|_p$. This is a version of Gauss's lemma.

Final note: the way that your problem is phrased, there is no bound, since if $G$ is a factor of $F$, then so is $cG$ for any nonzero constant $G$. In Lang's formulation, if you shift a constant from one factor to another, it doesn't affect the statement of the result.

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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.

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Right. And now that you mention it, Lang includes this in his book because "Michel Waldschmidt recommended their inclusion" (page 59). So Prop 2.12 on page 61 says that if $f,g\in\mathbb{C}[X_1,\ldots,X_n]$ and if $fg$ has total degree at most $d$, then $|f|_\infty|g|_\infty\le e^{nd}|fg|_\infty$. The proof, as you indicate, uses Mahler measure. Lang refers to Gelfond's book, as well as the article by Mahler that you quote. –  Joe Silverman Dec 14 '13 at 19:19

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