Soit $X$ une variété projective sur un corps de nombres $K$ (resp. sur $ℂ$). Soit $H$ la somme de « suffisamment de diviseurs positifs » sur $X$. On montre que tout ensemble de points quasi-entiers (resp. toute courbe entière) dans $X-H$ est non Zariski-dense.

The purpose of this article is twofold. The first is to find the dimension of the set of integral points off divisors in subgeneral position in a projective algebraic variety $V\subset {\mathbb{P}}_{k̅}^m$, where k is a number field. As consequences, the results of Ru-Wong (1991), Ru (1993), Noguchi-Winkelmann (2003) and Levin (2008) are recovered. The second is to show the complete hyperbolicity of the complement of divisors in subgeneral position in a projective algebraic variety $V\subset {\mathbb{P}}_{ℂ}^m.$

A number of authors have proven explicit versions of Lehmer’s conjecture for polynomials whose coefficients are all congruent to $1$ modulo $m$. We prove a similar result for polynomials $f\left(X\right)$ that are divisible in $(\mathbb{Z}/m\mathbb{Z})\left[X\right]$ by a polynomial of the form $1+X+\cdots +X^n$ for some $n\ge ϵdeg\left(f\right)$. We also formulate and prove an analogous statement for elliptic curves.