From Diophantine approximations to fundamental units of algebraic fields

Suppose we have $m$ real homogeneous forms $f_{i}(X)$, $i = 1,\ldots,m$ in the real $n$-dimensional space $\mathbb{R}^{n} = \{X\}$, $2\le m\le n$. In many cases, the convex hull of the set of points $G(X) = (|f_{1}(X)|,\ldots,|f_{m}(X)|)$ for $X\in \mathbb{Z}^{n}$ is a convex polyhedral set, and its boundary can be computed by means of the standard program for $\|X\|<\text{const}$. The boundary points $G(X)$, that is, the points lying on the boundary, correspond to the best Diophantine approximations $X$ for the above forms. This gives the global generalization of the continued fraction. For $n=3$, Euler, Jacobi, Dirichlet, Hermite, Poincare, Hurwitz, Klein, Minkowski, Brun, Arnold and lot of others tried to generalize the continued fraction, but without a success.


Let $p(\xi)$ be real polynomial of degree $n$, which is irreducible over $\mathbb{Q}$, and let $\lambda$ be its root. The set of fundamental units of the ring $\mathbb{Z}[\lambda]$ can be computed using the boundary points of some set of linear and quadratic forms constructed by means of the roots of the polynomial $p(\xi)$. Similarly, one can compute a set of fundamental units of the field $\mathbb{Q}(\lambda)$. Up today, such set of units was computed only for $n=2$ (using usual continued fractions) and $n=3$ (by the algorithm of Voronoi).

Our approach generalizes the continued fraction and gives the best Diophantine approximations and fundamental units of algebraic fields for any $n$.