Scientific computation on some mathematical problems

Our goal is to survey, using three examples, how high-precision computations have stimulated mathematical research in the areas of polynomial and rational approximation theory. The first example will be the “19” Conjecture in rational approximation theory. Here high-precision computations gave strong evidence that this conjecture is false. Gonchar and Rakhmanov have given an exact solution of this conjecture. The second example will be the “8” Conjecture in rational approximation theory. In this case, high-precision computations and the use of the Richardson extrapolation method led to this conjecture. Stahl has proved that this conjecture and its generalization are true. The final example will be the Bernstein Conjecture in polynomial approximation theory and its generalization.