Complex Zolotarev Polynomials on the Real Interval [−1, 1]

We consider complex Zolotarev polynomials of degree n on [−1, 1], i.e., monic polynomials of degree n with the second coefficient assigned to a given complex number ρ, that have minimum Chebyshev norm on [−1, 1]. They can be characterized either by n or by n+1 extremal points. We show that those corresponding to n extrema are closely related to real Zolotarev polynomials on [−1, 1], so that we distinguish between a trigonometric case where an explicit expression is given and the more complicated elliptic case. The classification of the parameters ρ that lead to one of the above cases is provided.