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 <n 1 on the compact interval [a, ft] the n-vector of its coefficients with respect to the power basis. It is shown that the condition number IIA^II^IIAÎ" 11^ 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 + \J2. In the more difficult case of an asymmetric interval around the origin we obtain upper bounds for the condition number which also grow exponentially.