The MO answer http://mathoverflow.net/a/98176/11926 notes the following: Let $\gamma$ be an algebraic number that is not a root of unity. Then Baker's theorem implies that there is a constant $C(\gamma)>0$ such that for all $n\ge2$ we have $|\gamma^n-1|\ge n^{-C(\gamma)}$. An alternative way to phrase this is that $\gamma$ cannot be too close to a root of unity. Thus if we let $\boldsymbol{\mu}_n$ denote the $n$'th roots of unity, then the result says that $\min_{\zeta\in\boldsymbol{\mu}_n}|\gamma-\zeta|\ge n^{-C(\gamma)}$. My question is whether a higher dimensional analogue of this is known. Let $F(X_1,\ldots,X_k)\in\overline{\mathbb{Q}}[X_1,\ldots,X_k]$ be a polynomial with algebraic coefficients. Does there exist a constant $C=C(F)>0$ such that $$ \min_{\substack{\zeta_1,\ldots,\zeta_k\in\boldsymbol{\mu}_n\\ F(\zeta_1,\ldots,\zeta_k)\ne0\\}} \bigl|F(\zeta_1,\ldots,\zeta_k)\bigr| \ge n^{-C} \quad\hbox{for all $n\ge2$?} $$ Or if this is not known, for the application that I have in mind it would suffice to have a lower bound which implies that $$ \liminf_{n\to\infty} \frac{1}{n} \min_{\substack{\zeta_1,\ldots,\zeta_k\in\boldsymbol{\mu}_n\\ F(\zeta_1,\ldots,\zeta_k)\ne0\\}} \log\bigl|F(\zeta_1,\ldots,\zeta_k)\bigr| \ge 0. $$

The MO answer https://mathoverflow.net/a/98176/11926 notes the following: Let $\gamma$ be an algebraic number that is not a root of unity. Then Baker's theorem implies that there is a constant $C(\gamma)>0$ such that for all $n\ge2$ we have $|\gamma^n-1|\ge n^{-C(\gamma)}$. An alternative way to phrase this is that $\gamma$ cannot be too close to a root of unity. Thus if we let $\boldsymbol{\mu}_n$ denote the $n$'th roots of unity, then the result says that $\min_{\zeta\in\boldsymbol{\mu}_n}|\gamma-\zeta|\ge n^{-C(\gamma)}$. My question is whether a higher dimensional analogue of this is known. Let $F(X_1,\ldots,X_k)\in\overline{\mathbb{Q}}[X_1,\ldots,X_k]$ be a polynomial with algebraic coefficients. Does there exist a constant $C=C(F)>0$ such that $$ \min_{\substack{\zeta_1,\ldots,\zeta_k\in\boldsymbol{\mu}_n\\ F(\zeta_1,\ldots,\zeta_k)\ne0\\}} \bigl|F(\zeta_1,\ldots,\zeta_k)\bigr| \ge n^{-C} \quad\hbox{for all $n\ge2$?} $$ Or if this is not known, for the application that I have in mind it would suffice to have a lower bound which implies that $$ \liminf_{n\to\infty} \frac{1}{n} \min_{\substack{\zeta_1,\ldots,\zeta_k\in\boldsymbol{\mu}_n\\ F(\zeta_1,\ldots,\zeta_k)\ne0\\}} \log\bigl|F(\zeta_1,\ldots,\zeta_k)\bigr| \ge 0. $$