OP-0025Partially solvedAnalysis

Sendov's conjecture

Tao proved it for polynomials of sufficiently large degree. The remaining degrees are open.

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Problem statement

Let $p$ be a polynomial of degree $n \ge 2$ with all roots in the closed unit disc, and let $a$ be one of its roots. Conjecture: the disc of radius $1$ centred at $a$ contains a critical point of $p$ , that is, a root of $p'$ .

Lean 4

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Formal statement

The formalization is available at the source below; its code is not attached here yet.

Formal Conjectures — Sendov's conjecture ↗

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A source attachment or link is not a verified proof. The statement’s correspondence with the original question needs human review.

Proofs & verification

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Background

Posed by Blagovest Sendov around 1958 and often misattributed to Ilieff. It is one of the most-attempted problems in the geometry of polynomials.

Known partial results

Proved for $n \le 8$ by a sequence of authors, and for various special configurations of roots.
Tao (2020) proved the conjecture for all sufficiently large $n$ , using a compactness and quantitative-perturbation argument. The proof is ineffective in the sense that it does not produce an explicit threshold.
What remains is the middle range: degrees above the verified small cases and below Tao's unspecified threshold.

References

[1]T. Tao, Sendov's conjecture for sufficiently high degree polynomials (2020). link
[2]Formal Conjectures — Sendov's conjecture link

Recorded progress

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Machine attempts

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Discussion

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