Grands degrés de transcendance pour des familles d'exponentielles

Abstract

We improve lower bounds obtained by P. Philippon for the transcendence degree over Q of families of the type {exp( u i v j ); i, j}, { v j , exp( u i v j ); i, j}. As a corollary we replace the lower bound [ d 2 ] by [ (d + 1) 2 ] for the family {exp( β j log α); 1 ≤ j ≤ d − 1} where α and β are algebraic and β of degree d ≥ 2; when d = 3 we give also a measure of algebraic independence of exp(β log α) and exp( β 2 log α).