On Sendov's Conjecture

Abstract

Sendov's Conjecture states that if a complex polynomial of degree greater than or equal to two has all its zeros inside the closed unit disk, then each zero is at distance no more than one from at least one critical point. This conjecture is known for degree n<9, but only partial results are available for higher n. In 2020, Prof. Terence Tao proved this conjecture for sufficiently high degree polynomials in a singular contribution that departs from conventional approaches. This thesis studies his work after examining the existing ground of theorems that relate the zeros and critical points of complex polynomials.