The condition of polynomials in power form

A study is made of the numerical condition of the coordinate map Mn which associates to each polynomial of degree 6 n 1 on the compact interval [a, b I the n-vector of its coefficients with respect to the power basis. It is shown that the condition number IIMnj IjMn Mlloo increases at an exponential rate if the interval [a, b] is symmetric or on one side of the origin, the rate of growth being at least equal to 1 + y/T. In the more difficult case of an asymmetric interval around the origin we obtain upper bounds for the condition number which also grow exponentially.