Hyperbolic automorphisms and Anosov diffeomorphisms on nilmanifolds

We translate the problem of finding Anosov diffeomorphisms on a nilmanifold which is covered by a free nilpotent Lie group into a problem of constructing matrices in $\mathrm{GL}(n,\mathbb{Z})$ whose eigenvalues satisfy certain conditions. Afterwards, we show how this translation can then be solved in some specific situations. The paper starts with a section on polynomial permutations of $\mathbb{Q}^K$, a subject which is of interest on its own.