AI Cracks Sendov’s Conjecture and Uncovers a Stronger Result Highlighted by Terence Tao

Recent advances in AI reasoning have accelerated progress on long‑standing mathematical problems. A 70‑year‑old open problem, Sendov’s conjecture, was proved with the help of GPT‑5.6 Pro, resulting in a paper titled “A Computer‑Assisted Proof of Sendov’s Conjecture” that contains roughly 90 000 lines of Lean 4 formal code.

Lech Mazur, the CEO of a startup and creator of the ProofAtlas platform, announced the result and provided a link to the PDF. Shortly after, Terence Tao posted a blog entry describing how he spent several days digesting, simplifying, and re‑formalising the proof, reducing the Lean code to about 15 000 lines and discovering that the argument actually proves a stronger statement, the 1972 Phelps‑Rodriguez conjecture.

Sendov’s Conjecture

The conjecture, posed by Bulgarian mathematician Blagovest Sendov in 1958, states that for any complex polynomial p(z) of degree n ≥ 2 whose zeros all lie in the closed unit disk, each zero a has at least one critical point w (a zero of p′) with |w − a| ≤ 1.

Equivalently, if all roots of a polynomial lie inside the unit circle, every root is within distance 1 of a critical point. The constant 1 is sharp, as shown by the polynomial p(z)=zⁿ−1.

Historical Progress

1969 – Meir and Sharma proved the case n < 6.

1991 – Brown extended to n < 7.

1996 – Borcea extended to n < 8.

1999 – Brown and Xiang extended to n < 9; no further low‑degree advances for two decades.

2020 – Terence Tao proved the conjecture for sufficiently large n (without an explicit bound).

Early 2026 – Teng Zhang made Tao’s bound explicit, up to 10^200 000.

Proof Outline (as presented by Tao)

Core assumption : suppose a counterexample exists; a degree‑n polynomial p has all zeros in the unit disk but some zero a lacks any critical point within distance 1.

Step 1 – Normalisation : rotate the plane so that a becomes a real number in [0,1). Replace each critical point w_j by its reciprocal coordinate q_j = 1/(a − w_j). The “no critical point within 1” condition translates into a bound on the q_j’s, packaging the alleged counterexample as two point sets inside the unit disk – the remaining zeros z_j and the reciprocal critical points q_j.

Step 2 – Communication identities : Tao introduces four algebraic relations (centroid identity, polarisation identity, first and second origin identities) obtained by evaluating p and p′ at natural points. These identities link the centroids of the zero set and the critical‑point set and provide algebraic constraints.

Step 3 – Branch point : Using the polarisation identity together with a Möbius‑transform estimate, Tao derives a crucial lower bound for an integral expression involving a. This is the only place where a is required to be real and marks the divergence of the argument.

Low‑degree case (n ≤ 5) : By defining a scalar function X(t)=a+(1−a²)t and analysing the integrand term‑by‑term, Tao shows the integral is strictly less than 1 for 0 < a < 1 and m ≤ 4, contradicting the lower bound.

High‑degree case (n ≥ 5) : Two inequalities are needed. The first comes from relaxing the integral via the AM‑GM inequality (the “polarisation inequality”). The second combines the first and second origin identities with the centroid identity (the “origin inequality”). These inequalities constrain the real part x of the average of the q_j’s and a parameter defined by the identities. For n ≥ 101 the two feasible regions are shown to be disjoint analytically; for 5 ≤ n ≤ 100 a computer‑checked rational‑coefficient Bernstein‑polynomial certificate verifies the incompatibility, with all checks performed in Lean.

Boundary case (|a| = 1) : Previously settled by Rubinstein, Tao reproduces the result using the same framework, replacing the polarisation identity with the Meir‑Sharma identity and forcing each q_j = 1, which forces p to be a scalar multiple of zⁿ − aⁿ. This precisely characterises the equality case excluded in the stronger Phelps‑Rodriguez conjecture.

Tao remarks that the proof is surprisingly elementary, relying only on the fundamental theorem of algebra, basic properties of Möbius transformations, and the Maclaurin inequality (derivable from the arithmetic‑geometric‑harmonic mean chain). The streamlined argument also yields the stronger Phelps‑Rodriguez conjecture, which requires the distance to be < 1 unless a lies on the unit circle and p is a scalar multiple of zⁿ − aⁿ.

AI’s Role in the Discovery

The original proof was generated with GPT‑5.6 Pro and formalised in Lean 4, producing about 90 000 lines of code. Mazur’s ProofAtlas platform visualises the proof graph, dependencies, and refutation paths. Tao’s subsequent work used AI assistance to digest and simplify the proof, reducing the formalisation to roughly 15 000 lines, which he open‑sourced on GitHub ( github.com/teorth/sendov).

This case illustrates a new model of human‑AI collaboration: AI assists in exploration, proof development, testing, and verification, while human mathematicians focus on judgment, synthesis, and connecting ideas. Formal verification via Lean provides a trustworthy foundation for AI‑generated proofs, as the type‑checker guarantees logical correctness of every step.

Despite the breakthrough, Tao notes many related conjectures (Borcea, Schmeisser, Smale problems) remain open, and his attempts to attack them with AI have not yet succeeded, underscoring that AI can prove theorems but also opens many new questions.