Independent Study
Collaboration with: Matthew King
Some of the most natural and beautiful functions studied in mathematics are polynomials. Each polynomial has a complexity, called its degree. Meanwhile, the complex numbers are a very important space in mathematics, often represented two dimensionally, as in the square below. It is a classical fact that a polynomial of degree n has n roots in the complex numbers. Below each purple dot represents the root of a polynomial of degree ~500. So there are 500 dots, but why do they make such a beautiful shape? This 'equidistribution' phenomenon is captured by the Erdos-Turan theorem, which says that if we look at the roots of typical polynomials of large degree, they will get closer and closer to forming a perfect centered circle of radius 1.
Citation: François G. Dorais (https://mathoverflow.net/users/2000/fran%c3%a7ois-g-dorais), Why do roots of polynomials tend to have absolute value close to 1?, URL (version: 2022-12-29): https://mathoverflow.net/q/182744
