Sendov's conjecture, also called Ilieff's conjecture, concerns the relationship between the roots and the critical points of a complex polynomial. It states that for a polynomial with all roots inside the closed unit disk, each root lies within distance 1 of at least one critical point of the polynomial. The conjecture has been proven for polynomials of degree less than 9 and, separately, for sufficiently large degree; as of August 2026 a further proof attributed to an AI system had been circulated and used by Terence Tao to establish a related generalization, the Phelps-Rodriguez conjecture.
Facts
StatementSendov's conjecture proposes that for a polynomial of degree at least two whose roots all lie in the closed unit disk, every root lies within distance one of some critical point of the polynomial. 1 Proposed Year Progress Toward ResolutionThe conjecture was proved for degree n less than 6 by Meir and Sharma in 1969, extended to n less than 7 by Brown in 1991, to n less than 8 by Borcea in 1996, and to n less than 9 by Brown and Xiang in 1999. Terence Tao proved it in 2020 for polynomials of sufficiently high degree. On 5 August 2026 a proof for the general case, produced with the assistance of OpenAI's GPT-5.6 Pro, was posted by Lech Mazur, and Terence Tao subsequently published an exposition verifying and simplifying the argument. Given how recent this claimed full resolution is, it is recorded here as reported by the cited source rather than treated as beyond question. 1 Prize Status
Prize Status (category)w-freetextdim2b-0926: category derived from a free-text property; original status/verification detail carried on the source property. Classification
Resolution Status Open Questions
Prize StatusNo prize is recorded.
No source consulted this pass names a monetary prize for this conjecture. Sources
1. Sendov's Conjecture (Wikipedia)
Wikimedia FoundationHistory section
The conjecture was first proposed by Blagovest Sendov in 1959; he described the conjecture to his colleague Nikola Obreshkov.
Lead section, conjecture statement
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.
Lead section and History section
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.
- Lead section
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