Rational approximations of the exponential function at rational points

A Conjecture on Rational Approximations to Rational Points

In this paper, we examine how well a rational point P on an algebraic variety X can be approximated by other rational points. We conjecture that if P lies on a rational curve, then the best approximations to P on X can be chosen to lie along a rational curve. We prove this conjecture for a wide range of examples, and for a great many more examples we deduce our conjecture from Vojta’s Main Conj...

Rational Interpolation of the Exponential Function

Let m, n be nonnegative integers and a<m+n) be a set of m + n + 1 real in~lation points (not necessarily distinct). Let Rm.n = P m.n / Qm.n be the unique rational function with degPm.n ::; m, deg Qm,n :$n, that in~lates e'" in the points of a<nr+n). Ifm = mv, n = nv with mv + nv --+ 00, and mv /nv --+ A as v --+ 00, and the sets a<nr+n) are uniformly bounded, we show that Pm.n(z) --+ ~/(l+)'), ...

Distribution of Rational Points: A Survey

Points at Rational Distance from the Vertices of a Unit Polygon

Rational Interpolation at Chebyshev points

The Lanczos method and its variants can be used to solve eeciently the rational interpolation problem. In this paper we present a suitable fast modiication of a general look-ahed version of the Lanczos process in order to deal with polynomials expressed in the Chebyshev orthogonal basis. The proposed approach is particularly suited for rational interpolation at Chebyshev points, that is, at the...