A proof of the Schinzel-Zassenhaus conjecture on polynomials

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 $\mathrm{\mid }z\mathrm{\mid }\le 2^{-1\mathrm{/}(4n)}$. We also obtain a relative version of this result over the compositum $$\mathbb{Q}$^{$\mathrm{a}\mathrm{b}$} \cdot $\mathbb{Q}$^{$\mathrm{t}\mathrm{.}$p}$ofallabelianandalltotally$p$-adic extensions of $\mathbb{Q}$, for any fixed prime~$p$, and apply it to prove a ${\mathbb{Q}}^{\mathrm{a}\mathrm{b}}\cdot {\mathbb{Q}}^{\mathrm{t}\mathrm{.}p}$-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 $p(X)\mathrm{/}(X^k-1{)}^m$, where $p(X)\in \mathbb{Q}[X]$, with a further application to existence of a small critical value for certain rational functions.