摘要：An astonishing fact was established by Lee A. Rubel (1981): there exists a fixed non-trivial fourth-order polynomial differential algebraic equation (DAE) such that for any positive continuous function φ on the reals, and for any positive continuous function ϵ(t), it has a C∞ solution with |y(t)−φ(t)|<ϵ(t) for all t. Lee A. Rubel provided an explicit example of such a polynomial DAE. Other examples of universal DAE have later been proposed by other authors. However, Rubel's DAE \emph{never} has a unique solution, even with a finite number of conditions of the form y(ki)(ai)=bi. The question whether one can require the solution that approximates φ to be the unique solution for a given initial data is a well known open problem [Rubel 1981, page 2], [Boshernitzan 1986, Conjecture 6.2]. In this article, we solve it and show that Rubel's statement holds for polynomial ordinary differential equations (ODEs), and since polynomial ODEs have a unique solution given an initial data, this positively answers Rubel's open problem. More precisely, we show that there exists a \textbf{fixed} polynomial ODE such that for any φ and ϵ(t) there exists some initial condition that yields a solution that is ϵ-close to φ at all times. In particular, the solution to the ODE is necessarily analytic, and we show that the initial condition is computable from the target function and error function.
关键词：Mathematics - Classical Analysis and ODEs;Computer Science - Computational Complexity;Computer Science - Logic in Computer Science;Electrical Engineering and Systems Science - Systems and Control;Mathematics - Dynamical Systems
