A proof of the Schinzel-Zassenhaus conjecture on polynomials

Abstract

We prove that if $P(X) \in \mathbb{Z}[X]$ is an integer polynomial of degree $n$ and having $P(0) = 1$, then either $P(X)$ is a product of cyclotomic polynomials, or else at least one of the complex roots of $P$ belongs to the disk $|z| \leq 2^{ - 1 / (4n) }$. We also obtain a relative version of this result over the compositum $\mathbb{Q}^{\mathrm{ab}} \cdot \mathbb{Q}^{\mathrm{t.}p}$ of all abelian and all totally $p$-adic extensions of $\mathbb{Q}$, for any fixed prime~$p$, and apply it to prove a $\mathbb{Q}^{\mathrm{ab}} \cdot \mathbb{Q}^{\mathrm{t.}p}$-relative canonical height lower bound on the multiplicative group. Another extension is given to a uniform positive height lower bound, inverse-proportional to the total number of singular points, on holonomic power series in $\mathbb{Q}[[X]]$ and not of the form $p(X) / (X^k-1)^m$, where $p(X) \in \mathbb{Q}[X]$, with a further application to existence of a small critical value for certain rational functions.