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Abstract
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Let n be a positive integer, let G be a group and consider $\mathbf{\nu}=(\nu_1,\dots,\nu_n)\in G^n$. We say that $\mathbf{\nu}$ is a multiplicatively dependent n--tuple if there exists a nonzero vector $(k_1,\dots,k_n)\in\mathbb{Z}^n$ for which $\nu^{k_1}_1\cdots\nu^ {k_n}_n=1.$ Given a finite extension K of $\mathbb Q$, we denote by $M_{n,K}(H)$ the number of multiplicatively dependent n--tuples of algebraic integers in $(K^*)^n$ with naive height up to H and we denote by $M^*_{n, K}(H)$ the number of multiplicatively dependent n--tuples of algebraic numbers in $(K^*)^n$ with naive height up to H.
In this talk, we shall discuss various estimates and asymptotic formulas for $M_{n,K}(H)$ and $M^*_{n, K}(H)$ with $H\rightarrow \infty$. For every $\nu$ in $(K^*)^n$ we define the multiplicative rank m of $\nu$ in the following way: if $\nu$ has a root of unity among its coordinates, then we set m=0. Otherwise we set m to be the largest integer with $1\leq m\leq n $ for which each set of m coordinates of $\mathbf\nu$ is a multiplicatively independent m--touple.
We also consider the sets $M_{n,K,m}(H)$ (resp. $M^*_{n,K,m}(H)$) defined as the number of multiplicatively dependent n--tuple of algebraic integers (resp. algebraic numbers) in $(K^*)^n,$ with multiplicative rank equal to m and naive height up to H considering the same issues in this case.
Finally we consider the problem of enumerating all multiplicatively dependent n--tuple of algebraic integers (resp. numbers) with fixed degree.
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