A digestion of the proof of Sendov’s conjecture

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A digestion of the proof of Sendov’s conjecture

Hacker News·A digestion of the proof of Sendov’s conjecture·29 days ago

#ukKuRhPN

#terrytao #conjecture #inequality #thus #unit #disk

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This post concerns the following conjecture of Sendov , as well as its strengthening by Phelps–Rodriguez : Conjecture 1 (Sendov’s conjecture) Let , and let be a degree polynomial with all zeroes in the unit disk. Then for every zero of , there exists a critical point of with . Conjecture 2 (Phelps–Rodriguez conjecture) Let , and let be a degree polynomial with all zeroes in the unit disk. Then for every zero of , there exists a critical point of with , unless is on the unit circle and is a scalar multiple of . By applying a rotation around the origin, we can normalize to be a real number with . From the work of Rubinstein , both conjectures were already established in the case, so one can restrict to the case. Both of these conjectures then follow from Conjecture 3 (Sendov’s conjecture in interior) Let . Let be a degree polynomial with all zeroes in the unit disk. Then if is a zero of , there exists a critical point of with .…

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A digestion of the proof of Sendov’s conjecture