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Mahler's method in several variables

By Boris Adamczewski

Also appears in collection : Diophantine approximation and transcendence / Approximation diophantienne et transcendance

Any algebraic (resp. linear) relation over the field of rational functions with algebraic coefficients between given analytic functions leads by specialization to algebraic (resp. linear) relations over the field of algebraic numbers between the values of these functions. Number theorists have long been interested in proving results going in the other direction. Though the converse result is known to be false in general, Mahler's method provides one of the few known instances where it essentially holds true. After the works of Nishioka, and more recently of Philippon, Faverjon and the speaker, the theory of Mahler functions in one variable is now rather well understood. In contrast, and despite the contributions of Mahler, Loxton and van der Poorten, Kubota, Masser, and Nishioka among others, the theory of Mahler functions in several variables remains much less developed. In this talk, I will discuss recent progresses concerning the case of regular singular systems, as well as possible applications of this theory. This is a joint work with Colin Faverjon.

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Citation data

  • DOI 10.24350/CIRM.V.19445203
  • Cite this video Adamczewski, Boris (12/09/2018). Mahler's method in several variables. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19445203
  • URL https://dx.doi.org/10.24350/CIRM.V.19445203

Bibliography

  • Adamczewski, B., & Faverjon, C. (2018). Mahler's method in several variables I: The theory of regular singular systems. <arXiv:1809.04823> - https://arxiv.org/abs/1809.04823
  • Adamczewski, B., & Faverjon, C. (2018). Mahler's method in several variables II: Applications to base change problems and finite automata. <arXiv:1809.04826> - https://arxiv.org/abs/1809.04826

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