From Diophantine approximations to Diophantine equations

A. D. Bruno

Keldysh Institute of Applied Mathematics of Russian Academy of Sciences, Moscow

Abstract: Let in the real $n$-dimensional space $\mathbb{R}^n=\{X\}$ be given $m$ real homogeneous forms $f_i(X)$, $i=1,\dotsc,m$, $2\leqslant m\leqslant n$. The convex hull of the set of points $G(X)=(|f_1(X)|,\dotsc,|f_m(X)|)$ for integer $X\in\mathbb Z^n$ in many cases is a convex polyhedral set. Its boundary for $||X||<\mathrm{const}$ can be computed by means of the standard program. The points $X\in\mathbb Z^n$ are called boundary points if $G(X)$ lay on the boundary. They correspond to the best Diophantine approximations $X$ for the given forms. That gives the global generalization of the continued fraction. For $n=3$ Euler, Jacobi, Dirichlet, Hermite, Poincaré, Hurwitz, Klein, Minkowski, Brun, Arnold and a lot of others tried to generalize the continued fraction, but without a succes.
Let $p(\xi)$ be an integer real irreducible in $\mathbb Q$ polynomial of the order $n$ and $\lambda$ be its root. The set of fundamental units of the ring $\mathbb Z[\lambda]$ can be computed using boundary points of some set of linear and quadratic forms, constructed by means of the roots of the polynomial $p(\xi)$. Similary one can compute a set of fundamental units of other rings of the field $\mathbb Q(\lambda)$. Up today such sets of fundamental units were computed only for $n=2$ (using usual continued fractions) and $n=3$ (using the Voronoi algorithms).
Our approach generalizes the continued fraction, gives the best rational simultaneous approximations, fundamental units of algebraic rings of the field $\mathbb Q(\lambda)$ and all solutions of a certain class of Diophantine equations for any $n$.
Bibliography: 16 titles.

Keywords: generalization of continued fraction, Diophantine approximations, set of fundamental units, fundamental domain, Diophantine equation.

UDC: 517.36

Received: 05.05.2016
Accepted: 12.09.2016

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