Sendov's conjecture

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In mathematics, Sendov's theorem (formally a conjecture), sometimes also called Ilieff's conjecture, concerns the relationship between the locations of roots and critical points of a polynomial function of a complex variable. It is named after Blagovest Sendov. The conjecture states that for a polynomial f ( z ) = ( z − r 1 ) ⋯ ( z − r n ) , n ≥ 2 , {\displaystyle f(z)=(z-r_{1})\cdots (z-r_{n}),\quad n\geq 2,} with all roots r1, ..., rn inside the closed unit disk, each of the n roots is at a distance no more than 1 from at least one critical point. The Gauss–Lucas theorem says that all of the critical points lie within the convex hull of the roots. It follows that the critical points must be within the unit disk, since the roots are. The conjecture was previously proven for n < 9 by Brown and Xiang and for n sufficiently large by Tao. There is also a generalisation by Dean Phelps and Rene S. Rodriguez: Zbl 0244.30001 On 5 August 2026, a proof of Sendov's conjecture generated using OpenAI's GPT-5.6 Pro was posted by Lech Mazur. Terence Tao subsequently published an exposition verifying and simplifying the argument, and extended the method to resolve the Phelps–Rodriguez conjecture.