Bounded Generation by semi-simple elements: quantitative results

Corvaja, Pietro; Demeio, Julian L.; Rapinchuk, Andrei S.; Ren, Jinbo; Zannier, Umberto M.

Comptes Rendus. Mathématique, Tome 360 (2022) no. G11, pp. 1249-1255.

Nous prouvons que pour un corps de nombres $F$, la distribution des points d’un ensemble $\Sigma \subset {𝔸}_F^n$ admettant une paramétrisation purement exponentielle, par exemple un ensemble de matrices bornément engendré par des éléments semi-simples (diagonalisables), est de taille au plus logarithmique lorsqu’il est ordonné par hauteur. Par conséquent, on obtient qu’un groupe linéaire $\Gamma \subset {\mathrm{GL}}_n\left(K\right)$ sur un corps $K$ de caractéristique zéro admet un paramétrisation purement exponentielle si et seulement s’il est de type fini et la composante connexe de sa clôture de Zariski est un tore. Nos résultats sont obtenus via une inégalité sur la hauteur des tuples minimaux d’un polynôme purement exponentiel. Un ingrédient clé de notre démonstration est une version forte par Evertse du théorème sur l’équation en $S$-unités.

We prove that for a number field $F$, the distribution of the points of a set $\Sigma \subset {𝔸}_F^n$ with a purely exponential parametrization, for example a set of matrices boundedly generated by semi-simple (diagonalizable) elements, is of at most logarithmic size when ordered by height. As a consequence, one obtains that a linear group $\Gamma \subset {\mathrm{GL}}_n\left(K\right)$ over a field $K$ of characteristic zero admits a purely exponential parametrization if and only if it is finitely generated and the connected component of its Zariski closure is a torus. Our results are obtained via a key inequality about the heights of minimal $m$-tuples for purely exponential parametrizations. One main ingredient of our proof is Evertse’s strengthening of the $S$-Unit Equation Theorem.

Classification : 11D75

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