FROM DIOPHANTINE APPROXIMATIONS TO DIOPHANTINE EQUATIONS

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, Poincar\'e, 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 \(n\).