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Phelps-Rodriguez Conjecture

The Phelps-Rodriguez conjecture (Phelps and Rodriguez 1972) is the strengthening of the Sendov conjecture asserting that for a polynomial p of polynomial degree n>=2 with all zeros in the closed unit disk, every zero a has a critical point zeta satisfying

|zeta-a|<1,

unless |a|=1 and p(z) is a nonzero scalar multiple of z^n-a^n.

In the exceptional case, all critical points equal 0 and their distance from a is 1. Tao (2026) showed that the AI-generated proof of the Sendov conjecture also establishes this stronger statement, and supplied a Lean formalization.

See also

Critical Point, Polynomial, Sendov Conjecture, Unit Disk

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References

Phelps, D. and Rodriguez, R. S. "Some Properties of Extremal Polynomials for the Ilieff Conjecture." Kodai Math. Sem. Rep. 24, 172-175, 1972. https://doi.org/10.2996/kmj/1138846519.Tao, T. "A Digestion of the Proof of Sendov's Conjecture." Aug. 12, 2026. https://terrytao.wordpress.com/2026/08/12/a-digestion-of-the-proof-of-sendovs-conjecture/.

Cite this as:

Weisstein, Eric W. "Phelps-Rodriguez Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Phelps-RodriguezConjecture.html

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