TITLE: Small values of a polynomial evaluated at roots of unity
QUESTION [8 upvotes]: 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
$$
  \liminf_{n\to\infty} 
  \frac{1}{n}  \min_{\substack{\zeta_1,\ldots,\zeta_k\in\boldsymbol{\mu}_n\\
           F(\zeta_1,\ldots,\zeta_k)\ne0\\}} 
           \log\bigl|F(\zeta_1,\ldots,\zeta_k)\bigr| \ge 0.
$$

REPLY [4 votes]: Here is a related, but different, problem, as mentioned by @Andreas. Let $f$ be a Laurent polynomial in $n$ variables with integer coefficients. Define $\log_0(t)$ to be $\log t$ if $t>0$ and 0 if $t=0$. Consider finite subgroups $F$ of the multiplicative $n$-torus and the Riemann sums of $\log_0 |f|$ over $F$ as $F$ becomes more and more uniformly distributed. Do these always converge to the integral of $\log_0 |f|$ over the torus with respect to Lebesgue measure (which is the log of the Mahler measure of $f$)? This question is motivated by a dynamical question about periodic points.
Concrete case: Let $f(x,y) = 3 - x - x^{-1} - y - y^{-1}$. The zero set of $f$ on the 2-torus is an oval containing exactly four roots on unity (the intersections of the oval with the coordinate axes), so there are no subtleties about how various finite $F$s can intersect this oval. And yet, as far as I know, whether or not there is convergence of the Riemann sums is open even in this simple case.