Sendov's Conjecture

A polynomial is a formula that only adds and multiplies numbers and variables, such as $p (x) = x^{2} - 1$ . A zero of $p$ is an input that gives the output 0. Here the zeros are $1$ and $- 1$ . The graph of this $p$ is a U shape, and it is flat at its lowest point, $x = 0$ . A flat point is called a critical point.

The slope of the graph has its own formula, called the derivative and written $p^{'}$ . Here $p^{'} (x) = 2 x$ . The critical points are the zeros of $p^{'}$ . A curve must turn around between two zeros, so a critical point lies between them. So zeros and critical points are tied to each other.

Sendov's question uses complex numbers. A complex number has the form $a + b i$ , where $i^{2} = - 1$ . It is a point on a flat map, with $a$ across and $b$ up. On this map, a polynomial with highest power $n$ has $n$ zeros and $n - 1$ critical points, counted with repeats. The unit disk is the set of points within distance 1 of the origin.

A polynomial with three zeros (solid dots) in the unit disk, and its two critical points (open dots). The dashed circle has radius 1 around the zero at 1, and it holds a critical point.

Sendov's conjecture says this. If all the zeros of a polynomial lie in the unit disk, then each zero has a critical point within distance 1 of it. Any two points in the unit disk are at most 2 apart, so the limit of 1 is a real restriction.

The limit of 1 cannot go lower. Take $p (z) = z^{n} - 1$ . Its zeros are $n$ evenly spaced points on the edge of the unit disk. Its derivative is $n z^{n - 1}$ , so its only critical point is the origin. So each zero is exactly 1 from the nearest critical point.

An older theorem says more about where critical points lie. Stretch a loop around all the zeros, and every critical point lies inside that loop. This does not settle the question, because a zero at a far corner could still be far from all of them. Proofs existed for polynomials with highest power up to 8, and for very large powers, but not for the range between.

Terence Tao's account reports that Lech Mazur used an AI tool to prove the conjecture for all powers, with a proof checked in Lean. Lean is a computer program that checks each step of a proof. Tao then simplified the argument, and Tao's Lean version has about 15,000 lines, against about 90,000 before.

Tao reports that the proof uses little beyond the fact that a polynomial with highest power $n$ has $n$ zeros. It also gives a stronger result. The distance reaches exactly 1 only for a multiple of $z^{n} - a^{n}$ , with the zero $a$ on the edge of the disk.