Numerical Approximation of Real Functions and One Minkowski's Conjecture on Diophintine Approximations

Abstract

In this paper I consider the applications of several kinds of approximations of real functions to the problem of verified computation (reliable computing) of the range of implicitly defined real function $x_{n+1} = G(x_{1}, ..., x_{n}),$ where dependency $F(x_{1}, ..., x_{n+1}) = 0$ is defined on some compact domain by a sufficiently smooth real function $F(x_{1}, ..., x_{n+1}) >.$ Constructive version of Kolmogorov-Arnold and implicit function theorems, results about floating-point approximation, floating-point approximations which give lower-bound and upper-bound estimates of some real functions, and approximate algebraic computation are used for the purpose. The rigorous theory can be build on the base of analysis on manifolds over floating points domains. In the text we demonstrate our approach on examples from investigation of Minkowski's conjecture on critical determinant of the region $\mid x \mid^p + \mid y \mid^p \leq 1, p > 1.$.