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Algebraic Independence in Algebraic Groups. Part II: Large Transcendence Degrees

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1752)

Abstract

This chapter is a continuation of chapter 13 on small transcendence degrees.

Our first goal is to introduce conjectures. We are not looking for the most general ones (see [And2]: we only propose some open problems which might be easier to prove, given the currently available methods.

Our next purpose is to present a proof of a result of algebraic independence which shows that some fields have a “large transcendence degree”: these fields are generated by numbers of the form αβ, αβ 2,..., when α and β are algebraic. We do not give the proof of the best known result on this topic (due to G. Diaz [Dia2], but only of a weaker statement for which the arguments may look more transparent. We shall explain how to use the transcendence criterion (chapter 8) and the zero estimate (chapter 11) together with an auxiliary function.

Keywords

  • Algebraic Group
  • Positive Real Number
  • Total Degree
  • Linear Algebraic Group
  • Transcendence Degree

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© 2001 Springer-Verlag Berlin Heidelberg

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(2001). Algebraic Independence in Algebraic Groups. Part II: Large Transcendence Degrees. In: Nesterenko, Y.V., Philippon, P. (eds) Introduction to Algebraic Independence Theory. Lecture Notes in Mathematics, vol 1752. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44550-1_14

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  • DOI: https://doi.org/10.1007/3-540-44550-1_14

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-41496-4

  • Online ISBN: 978-3-540-44550-0

  • eBook Packages: Springer Book Archive