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In 1985, Yu. V. Nesterenko produced a criterion for linear independence,which is a variant of Siegel's. While Siegel uses upper bounds on full systemsof forms, Nesterenko uses upper and lower bounds on sufficiently densesequences of individual forms. The proof of Nesterenko's criterion wassimplified by F. Amoroso and P. Colmez in 2003. More recently, S. Fischler andW. Zudilin produced a refinement, together with a much simpler proof. This newproof rests on a simple argument which we expand here. We get a new result,which contains Nesterenko's criterion, as well as criteria for algebraicindependence.

On the criteria for linear independence of Nesterenko, Fischler and Zudilin