12
OCT
2020

Vesselin Dimitrov told about his proof of the Schinzel--Zassenhaus conjecture

Today at the number theory seminar Veselin Dimitrov from the University of Toronto was speaking about his proof of the Schinzel--Zassenhus conjecture. The conjecture describes behaviour of complex roots of polynomials with integer coefficients. In 1960s Andrzej Schinzel and Hans Zassenhaus predicted that either all roots are on the unit circle (such polynomials are called cyclotomic) or there is a root at some mimimum distance away, |z| > 1 + C/n, where n is the degree of the polynomial and C is an absolute constant. At the end of 2019 Dimitrov gave a proof, which combined ingenious elementary observations and deep knowledge of properties of analytic functions. In the spring professor Dimitrov accepted the invitation to give a lecture at IM PAN, but it was later cancelled due to the pandemics. As the number theory seminar at the Institute resumed, professor Dimitrov kindly offered us a remote lecture. He outlines the proof of the conjecture, discusses other possible applications of his methods and states some open problems. The record of the talk is available here.