Abstract
We prove the Schinzel–Zassenhaus conjecture, in the following explicit form. If $P(X) \in \mathbf{Z}[X]$ 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 disc $|X| \leq 2^{ – 1 / (4n) }$. We also obtain a relative version of this result over the compositum $\mathbf{Q}^{\mathrm{ab}} \cdot \mathbf{Q}^{\mathrm{t}.p}$ of all abelian and all totally $p$-adic extensions of $\mathbf{Q}$, for any fixed prime $p$, and apply it to prove a $\mathbf{Q}^{\mathrm{ab}} \cdot \mathbf{Q}^{\mathrm{t}.p}$-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 $\mathbf{Q} [[X ]]$ and not of the form $p(X) / (X^l-1)^m$, where $p(X) \in \mathbf{Q}[X]$. We derive applications to upper bounds on the number of noncyclotomic irreducible factors of a lacunary polynomial, and a further application to the existence of a small critical value for certain rational functions.
