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Algebraic Independence Measures for Values of E-functions and M-functions

Colin Faverjon, Boris Adamczewski

TL;DR

The paper develops Liouville-type lower bounds for polynomials evaluated at the values of Siegel $E$-functions and Mahler $M$-functions at non-zero algebraic points, removing key regularity and independence assumptions. It combines Siegel–Shidlovskii techniques with elimination theory, Padé approximants, and Hilbert-Serre arguments to produce effective lower bounds of the form $|P(f_1(\alpha),\ldots,f_m(\alpha))| \ge C_1 H(P)^{-C_2 \deg(P)^t}$ (up to the alternative that the value vanishes), where $t$ is the transcendence degree. The results generalize previous works by not requiring full independence or a fixed-size system, and they yield consequences for Mahler’s classification, notably excluding Liouville-type numbers from the sets of values $f E\uplus\bf M$. The approach also yields an independent proof that no $f E\uplus\bf M$ value is a Liouville number, and provides effective constants and explicit regime-based bounds, with extensions to the sans-system setting via regular differential/Mahler operators.

Abstract

In this article, we establish a Liouville-type inequality for polynomials evaluated at the values of arbitrary Siegel E-functions at non-zero algebraic points. Additionally, we provide a comparable result within the framework of Mahler M -functions.

Algebraic Independence Measures for Values of E-functions and M-functions

TL;DR

The paper develops Liouville-type lower bounds for polynomials evaluated at the values of Siegel -functions and Mahler -functions at non-zero algebraic points, removing key regularity and independence assumptions. It combines Siegel–Shidlovskii techniques with elimination theory, Padé approximants, and Hilbert-Serre arguments to produce effective lower bounds of the form (up to the alternative that the value vanishes), where is the transcendence degree. The results generalize previous works by not requiring full independence or a fixed-size system, and they yield consequences for Mahler’s classification, notably excluding Liouville-type numbers from the sets of values . The approach also yields an independent proof that no value is a Liouville number, and provides effective constants and explicit regime-based bounds, with extensions to the sans-system setting via regular differential/Mahler operators.

Abstract

In this article, we establish a Liouville-type inequality for polynomials evaluated at the values of arbitrary Siegel E-functions at non-zero algebraic points. Additionally, we provide a comparable result within the framework of Mahler M -functions.

Paper Structure

This paper contains 28 sections, 13 theorems, 103 equations.

Table of Contents

  1. Introduction
  2. Organization of the Paper
  3. Main definitions
  4. $E$-functions
  5. $M$-functions
  6. Mahler's classification
  7. Proof of Case (E) of Theorem \ref{['thm:measure_system']}
  8. Preliminary results on dimensions of some vector spaces
  9. A first basis of linear forms
  10. Linear forms coming from the functional relations
  11. Linear forms coming from Siegel's method
  12. Linear forms coming from $\mathcal{J}$
  13. Construction of $L_{q+1},\ldots,L_r$
  14. Construction of $N_1,\ldots,N_u$
  15. End of the proof of Case (E) of Theorem \ref{['thm:measure_system']}
  16. ...and 13 more sections

Key Result

Theorem A

Let $m\geq 1$ be an integer, $f_1,\ldots,f_m \in \overline{\mathbb Q}[[z]]$ be algebraically independent over $\overline{\mathbb Q}(z)$ and $\alpha \in \overline{\mathbb Q}\setminus\{0\}$. Then, in the two following cases, there exist two positive real numbers $C_1$ and $C_2$ such that the inequalit holds for all non-zero polynomials $P\in \mathbb Z[X_1,\ldots,X_m]$. In each case, $C_1$ depends o

Theorems & Definitions (33)

  • Theorem A
  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Corollary 1.5
  • Conjecture 2.1
  • Lemma 3.1
  • proof
  • Remark 3.2
  • ...and 23 more