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A maximal polynomial is a complex polynomial that has all of its roots in the unit disk, one fixed root, and all of its critical points as far as possible from a fixed point. In this paper we determine a lower bound for the number of roots and critical points of a maximal polynomial that must lie on specified circles.doi:10.1090/s0002-9939-1992-1113647-x fatcat:xre37fwc35awtod6fmflogn7qa