Application of Quantifier Elimination to Solotareff’s Approximation Problem

Abstract

Solotareff’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 generality we take the approximated polynomial to be x ⁿ + rx ⁿ −¹,r ≥ 0. We treat r as a parameter and seek to compute the coefficients of the best approximations, for small fixed values of n, as piecewise algebraic functions of r. We succeed only for n ≤ 4, but also find that we can easily compute the coefficients for n = 5 for fixed values of r. The results serve to display the capabilities and limitations of our quantifier elimination program qepcad. We also prove that the coefficients of the best approximation, for any n, are continuous functions of r.

Supported by the Austrian Science Foundation (Grant P8572-PHY).