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Multiplicity estimates for algebraically dependent analytic functions

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Publication details

JournalProceedings of the London Mathematical Society
DateE-pub ahead of print - 8 Oct 2013
DatePublished (current) - Apr 2014
Issue number4
Number of pages41
Pages (from-to)989-1029
Early online date8/10/13
Original languageEnglish


We prove a new general multiplicity estimate applicable to sets of functions without any assumption on algebraic independence. The multiplicity estimates are commonly used in determining measures of algebraic independence of values of functions, for instance, within the context of Mahler’s method. For this reason, our result provides an important tool for the proofs of algebraic independence of complex numbers. At the same time, these estimates can be considered as a measure of algebraic independence of functions themselves. Hence, our result
provides, under some conditions, the measure of algebraic independence of elements in Fq[[T]], where Fq denotes a finite field.

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