Zero Estimates on Commutative Algebraic Groups

Abstract

Zero estimates appear as a key stone in the theory of transcendental numbers. The construction of an auxiliary function, by analytic means, typically produces a polynomial vanishing at many points of an algebraic group. The object of a zero estimate is to extract as much information as possible from these data. We shall discuss here a zero estimate due to P. Philippon [PPh3], which contains earlier work by D. W. Masser [Mas2], by D. W. Masser and G. Wüstholz [MW1], [MW3], and by G. Wüstholz [Wus1], [Wus2]. For further study, the reader is encouraged to look at the more recent works of L. Denis [Den], M. Nakamaye [Nak] and P. Philippon [PPh9]. The reader may also look at the paper of J.-C. Moreau [Mor] which gives a geometric exposition of the zero estimate of [MW1], together with a generalization to multi-projective space. Another exposition of zero estimates including their applications to transcendental number theory can be found in the Bourbaki lecture of D. Bertrand [Ber5].

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