Géométrie algébrique, Théorie des nombres

Bounded Generation by semi-simple elements: quantitative results

[Engendrement borné par éléments semi-simples : résultats quantitatifs]

Corvaja, Pietro¹  ; Demeio, Julian L.²  ; Rapinchuk, Andrei S.³  ; Ren, Jinbo⁴  ; Zannier, Umberto M.⁵ 

¹ Dipartimento di Scienze Matematiche, Informatiche e Fisiche, via delle Scienze, 206, 33100 Udine, Italy
² Departement Mathematik und Informatik, Universität Basel, 4051 Basel, Switzerland
³ Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USA
⁴ School of Mathematical Sciences, Xiamen University, Xiamen 361005, China
⁵ Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa, Italy

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

Abstract (VO)

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.

Résumé

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.

Détail

Reçu le : 2022-03-01
Révisé le : 2022-04-12
Accepté le : 2022-05-11
Publié le : 2022-12-08

DOI : 10.5802/crmath.376

Classification : 11D75

Licence : CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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