I said
For a case like x^N, the Remez equations are all algebraic. Minimaxing absolute error in the cubic approximating x^4 on [0,1], I got a mess of complex solutions, plus two reals. One was x^4-T[4](x)/8 as you'd expect, since Chebychevs are equal-ripple approximations to 0. This turned out minimax on [-1,1], with amplitude |mu| = 1/8, and extrema -1,-1/sqrt2, 0,1/sqrt(2), and 1. But much better (on [0,1]) was x^4 ~ 2x^3 - 5x^2/4 + x/4 - 1/128, with |mu| = 1/128 and extrema 0, (2-sqrt2)/4, 1/2, (2+sqrt2)/4, and 1.
Privately, Rich Fateman said
Chebyshev approximations can be used on any interval by a linear transformation.
Indeed, 2x^3 - 5x^2/4 + x/4 - 1/128 is just x^4-T[4](2x-1)/128 . But Chebychevs only work once. I.e., you can't Tcheb off the 2x^3 term and get an equal-ripple quadratic approximation to x^4. Nor can you introduce weight functions, nor relative errors. But Tchebs are useful to Remezing in two different ways: Their turning points are good first approximations to Remez turning points, and likewise their cofficients, either for initializing the iterative algorithm, or the direct Newton's attack on the whole system. --rwg PAST DUE UPDATES (speaking of which: Is there really no PolynomialDiscriminant in Mma? It's just Resultant[p,p',...].)