Application of Quantiier Elimination to Solotaree"s Approximation Problem

Solotaree's approximation problem is that of obtaining the best uniform approximation on the interval-1,+1] of a real polynomial of degree n by one of degree n ? 2 or less. Without loss of generalitywe take the approximated polynomial to be x n +rx n?1 , r 0. We treat r as a parameter and seek to compute the coeecients of the best approximations, for small xed values of n, as piecewise algebraic functions of r. We succeed only for n 4, but also nd that we can easily compute the coeecients for n = 5 for xed values of r. The results serve to display the capabilities and limitations of our quantiier elimination program qepcad. We also prove that the coeecients of the best approximation, for any n, are continuous functions of r.