Let S_n be the set of monic complex polynomials of degree n≥ 2 whose zeros lie in the closed unit disk. For p∈ S_n, define d(p)=max_z∈ Z(p)min_w∈ Z(p')|z-w|. Call p extremal for Sendov's conjecture if d(p)=sup_q∈ S_nd(q). Phelps–Rodriguez conjecture. If p∈ S_n is extremal for Sendov's conjecture, then p(z)=z^n+e^i theta for some theta∈mathbfR. This is a strengthened form of Sendov's conjecture. The paper describes it as a conjecture proposed by Phelps and Rodriguez; its general status is not resolved here.

Phelps–Rodriguez conjecture on extremal Sendov polynomials

From papers

Let $S_n$ be the set of monic complex polynomials of degree $n\ge 2$ whose zeros lie in the closed unit disk. For $p\in S_n$ , define

d(p)=\max_{z\in Z(p)}\min_{w\in Z(p')}|z-w|.

Call $p$ extremal for Sendov's conjecture if $d(p)=\sup_{q\in S_n}d(q)$ . Phelps–Rodriguez conjecture. If $p\in S_n$ is extremal for Sendov's conjecture, then

p(z)=z^n+e^{i\theta}

for some $\theta\in\mathbf{R}$ . This is a strengthened form of Sendov's conjecture. The paper describes it as a conjecture proposed by Phelps and Rodriguez; its general status is not resolved here.

Progress summary

Solved

A new exposition claims a fully checked proof settles this conjecture, but independent confirmation of the strengthened claim has not appeared.

Phelps and Rodriguez proposed this strengthening of Sendov’s conjecture in 1972: every extremal polynomial should be of the form $p(z)=z^n+e^{i\theta}$ . The assertion concerns monic degree- $n$ polynomials whose zeros lie in the unit disk.

Known results

Rubinstein proved a related Phelps–Rodriguez statement for polynomials with a prescribed zero.
Bojanov, Rahman, and Szynal proved $d(P,0)<1$ for every $P\in S(0)$ .
For the restricted class $S(n,0)$ , $d(p)\leq\left(\frac{1}{n}\right)^{\frac{1}{n-1}}$ , with equality only for the conjectured special polynomials.

August 2026 claimed resolution

On August 12, 2026, Terence Tao reported that a proof of the relevant conjecture establishes both Sendov’s conjecture and the Phelps–Rodriguez conjecture in full generality. He attributed the underlying work to Lech Mazur, who used an unnamed AI tool, and said the proof was verified in Lean; no independent verification of the Phelps–Rodriguez conclusion was found.

Current status (as of August 2026): The conjecture has a reported general proof with Lean verification of the underlying Sendov proof, but the strengthened Phelps–Rodriguez conclusion remains unconfirmed independently.

Sources

Sources & referencesView supporting material

Primary source

Julius Borcea, “Maximal and linearly inextensible polynomials”, arXiv:math/0601600 (2006).

Additional references

2 papers in this index state this conjecture (2003–2006). The statement above is taken from the most recent of them; the others are arXiv:math/0309233.

complex-analysis polynomial-zeroes

1 week ago