On Arithmetic Properties of the Solutions of a Universal Differential Equation at Algebraic Points Original Research Article

In 1981, L. A. Rubel found an explicit algebraic differential equation (ADE) of order four such that every real continuous function on the real line can be uniformly approximated by the C∞-solutions of this ADE. It is shown that an ADE of order five exists, where the C∞-solutions additionally satisfy some algebraic properties in the sense of C. L. Siegelʹs results from the analytical theory of numbers. For instance, all the solutions and their derivatives are transcendental at algebraic points, and large sets of these numbers are linearly independent over the field of real algebraic numbers.