We prove that Sendov's conjecture is true for polynomials of degree N = 7 and determine the so-called extremal polynomials in this case. As a consequence of a general result concerning polynomials with a prescribed number of distinct zeros (Theorem 2.4), we also check the validity of the conjecture for polynomials with at most seven distinct roots. Some properties of the extremal polynomials are discussed as well. 1991 MATH. SUBJ.
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