Computation of Zolotarev rational functions

Abstract: An algorithm is presented to compute Zolotarev rational functions, that is, rational functions $r_n^*$ of a given degree that are as small as possible on one set $E\subseteq\complex\cup{\infty}$ relative to their size on another set $F\subseteq\complex\cup{\infty}$ (the third Zolotarev problem). Along the way we also approximate the sign function relative to $E$ and $F$ (the fourth Zolotarev problem).