Sendov’s conjecture
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Sendov’s conjecture says that if every zero of a complex polynomial lies in the closed unit disk, then each zero is within distance one of a zero of its derivative. The target includes repeated and boundary zeros.
Why it matters and what counts as progress
Why it matters: A proof must cover the degree ranges left between earlier low-degree and sufficiently-high-degree results.
What counts as progress: A proof for every degree at least two, a newly covered degree range, or a rigorously verified zero-to-critical-point counterexample.
How the problem developed
- 1958 · Conjecture [1]
Blagovest Sendov Posed the conjecture that every zero of a degree-n polynomial in the unit disk lies within distance one of a critical point.
- 8 Dec 2020 · Partial result [1]
Terence Tao Proved the conjecture for all sufficiently large degrees, leaving only the intermediate degrees unresolved.
- 5 Aug 2026 · Proof [2]
Lech Mazur, using AI assistance Posted a Lean-verified proof resolving Sendov’s conjecture for every degree n≥2.
- 12 Aug 2026 · Exposition [3]
Terence Tao, with heavy AI assistance Published a digestion of Mazur’s proof and showed that its argument also proves the Phelps–Rodriguez strengthening.
AI contribution
A Computer-Assisted Proof of Sendov’s Conjecture
For every complex polynomial of degree at least two with all zeros in the closed unit disk, each zero lies within distance one of a zero of its derivative, including repeated and boundary zeros.
Sources
Website timeline
Publication dates and editorial changes.
AI-assisted proof covers Sendov’s conjecture in every degree