The Schinzel-Zassenhaus conjecture on integer polynomials is true

You need to know: Polynomials, degree of a polynomial, constant polynomial (the one of degree $0$ ), leading coefficient of a polynomial (the nonzero coefficient of highest degree), monic polynomial (polynomial with leading coefficient $1$ ), roots of a polynomial, set ${\mathbb C}$ of complex numbers, absolute value $|z|$ of complex numbers $z\in{\mathbb C}$ .

Background: Let ${\mathbb Z}[x]$ be the set of polynomials in one variable x with integer coefficients. Polynomial $Q(x) \in {\mathbb Z}[x]$ is called a divisor of $P(x) \in {\mathbb Z}[x]$ if $P(x)=Q(x)R(x)$ for some $R(x)\in {\mathbb Z}[x]$ . If $P(x) \in {\mathbb Z}[x]$ cannot be written as $P(x)=Q(x)R(x)$ for non-constant $Q(x),R(x) \in {\mathbb Z}[x]$ , we say that $P(x)$ is irreducible. An irreducible polynomial $P \in {\mathbb Z}[x]$ is called cyclotomic if $P$ is a divisor of $x^n-1$ for some integer $n \geq 1$ .

The Theorem: On 28th December 2019, Vesselin Dimitrov submitted to arxiv a paper in which he proved that every non-cyclotomic monic irreducible polynomial $P(x) \in {\mathbb Z}[x]$ of degree $n>1$ has at least one root $\alpha\in{\mathbb C}$ satisfying $|\alpha|\geq 2^{1/4n}$ .

Short context: It is easy to see that all roots of cyclotomic polynomials have absolute value 1. In 1965, Schinzel and Zassenhaus conjectured that any other monic irreducible polynomial of degree n must have a root $\alpha$ satisfying $|\alpha|\geq 1+\frac{c}{n}$ , where $c>0$ is a universal constant. The conjecture attracted a lot of attension, but, before 2019, was proved only in special cases, e.g. for polynomials with odd coefficients, as a corollary of this Theorem. Because $2^{1/4n} \geq 1+\frac{\log 2}{4n}$ , the Theorem confirms this conjecture in full generality, with $c=\frac{\log 2}{4}$ .