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Mahler's method and Carlitz logarithm

Guillaume Estienne

Abstract

In 2007, Papanikolas established that if Carlitz logarithms of algebraic functions are linearly independent over the rational function field, then they are algebraically independent. The purpose of the present paper is to provide a new proof of this theorem using Mahler s method instead of the theory of t-motives. We revisit and extend the approach developed by Denis, which enabled him in 2006 to prove this result in the particular case of the logarithm of elements in Fq(theta) via a Mahler system.

Mahler's method and Carlitz logarithm

Abstract

In 2007, Papanikolas established that if Carlitz logarithms of algebraic functions are linearly independent over the rational function field, then they are algebraically independent. The purpose of the present paper is to provide a new proof of this theorem using Mahler s method instead of the theory of t-motives. We revisit and extend the approach developed by Denis, which enabled him in 2006 to prove this result in the particular case of the logarithm of elements in Fq(theta) via a Mahler system.
Paper Structure (13 sections, 29 theorems, 131 equations)

This paper contains 13 sections, 29 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments:
  3. Algebraic Independence of logarithms
  4. Construction of the interpolation functions
  5. Functional equation
  6. Mahler system
  7. Preliminary constructions
  8. The main result
  9. Application
  10. Appendix: Proof of theorem \ref{['Adam_Denis']}
  11. An algebraic independence criterion
  12. Two preliminary lemmas
  13. Proof of the theorem

Key Result

Theorem 1.1

(Papanikolas 2007.) Let $\lambda_1,...,\lambda_n \in \mathbb{C}_\infty$ satisfy $\exp_C(\lambda_i) \in \overline{k}$ for $i \in [\![1,n]\!]$. If the family $( \lambda_i)_{i=1}^n$ is $k$-linearly independent, then it is $\overline{k}$-algebraically independent.

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 2.1
  • proof
  • Lemma 2.2
  • proof
  • Proposition 2.3
  • ...and 41 more