
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.
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UDC:
517.36 Received: 05.05.2016 Accepted:12.09.2016
Citation:
A. D. Bruno, “From Diophantine approximations to Diophantine equations”, Chebyshevskii Sb., 17:3 (2016), 38–52
Citation in format AMSBIB
\Bibitem{Bru16}
\by A.~D.~Bruno
\paper From Diophantine approximations to Diophantine equations
\jour Chebyshevskii Sb.
\yr 2016
\vol 17
\issue 3
\pages 3852
\mathnet{http://mi.mathnet.ru/cheb496}
\elib{http://elibrary.ru/item.asp?id=27452081}
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