Hope someone will help me in the solution of the following question. I'm working on some topological problem involving the topology of the intersection of some characters of the torus. I want ot find a way to exstimate the dimension of a manifold using some matrix. Here's the question; probably it's a well known problem in the literature
Let $n\geq1$ and let $\mathcal{W}\subseteq(S^{1})^{n}$ be the space of solution of the system $$z_{1}^{w_{1}^{1}}\cdots z_{n}^{{w_{1}^{n}}}=b_{1}$$ $$\cdots$$ $$z_{1}^{w_{m}^{1}}\cdots z_{n}^{{w_{m}^{n}}}=b_{m}$$
where $m\geq1,$ $w_{i}^{j}\in\mathbb{Z}$ for $1\leq i \leq m$ and $1\leq j\leq n$ and $b_{l}\in S^{1}$ for $1\leq l \leq m.$ Let $W\in M_{m,n}(\mathbb{Z})$ be the matrix of $m$ rows and $n$ columns defined by $$W=(w_{i}^{j})$$ Then, $\mathcal{W}$ is either empty or a finite disjoint union of subtori of $(S^{1})^{n},$ each of dimension $\operatorname{dim}\ker(W).$