Points at Rational Distances on a Parabola

Points on y = x2 at rational distance

Nathaniel Dean asks the following: Is it possible to find four nonconcyclic points on the parabola y = x2 such that each of the six distances between pairs of points is rational? We demonstrate that there is a correspondence between all rational points satisfying this condition and orbits under a particular group action of rational points on a fiber product of (three copies of) an elliptic surf...

Distribution of Rational Points: A Survey

POINTS ON y = x AT RATIONAL DISTANCE

Nathaniel Dean asks the following: Is it possible to find four nonconcyclic points on the parabola y = x2 such that each of the six distances between pairs of points is rational? We demonstrate that there is a correspondence between all rational points satisfying this condition and orbits under a particular group action of rational points on a fiber product of (three copies of) an elliptic surf...

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...