Grands degrés de transcendance pour des familles d'exponentielles. (Large transcendence degrees for families of exponentials) (Q1112864)

Consider families of the type \(\{\exp (u_ i v_ j);i,j\}\) and \(\{v_ j,\exp (u_ i v_ j);i,j\}\). The author improves the lower bounds obtained by P. Philippon for the transcendence degree of such families when some technical hypothesis are satisfied. As a consequence he narrows the Gel'fond-Schneider conjecture by obtaining the new lower bound \((d+1)/2\) instead of d/2 for a family \(\exp (\beta^ j \log \alpha)\), \(1\leq j\leq d-1\) (d\(\geq 2)\) with \(\alpha\) and \(\beta\) algebraic and \(\beta\) of degree \(d\geq 2.\) The author uses a one variable auxiliary function and Philippon's results that he sometimes improves, particularly a criterion of algebraic independence. The extension of these results to p-adic numbers seems to be likely and should be studied.