Algebraic independence of the values at algebraic points of a class of functions considered by Mahler

N. Ch. Wass

GDML_Books, (1990), p.

Résumé

This thesis is concerned with the problem of determining a measure of algebraic independence for a particular m-tuple θ₁,...,$θ_m$ of complex numbers. Specifically, let K be a number field and let f₁(z),...,$f_m(z)$ be elements of K[[z]] algebraically independent over K(z) satisfying equations of the form(*) $f_j(z^b) = ∑^m_{i=1} f_i(z)a_{ij}(z) + b_j(z)$ (j = i,...,m)for b ≥ 2, $a_{ij}(z)$, $b_j(z)$ in K(z). Suppose finally that α ∈ K is such that 0 < |α| < 1, the $f_j(z$) converge at z = α and the $a_{ij}(z)$, $b_j(z)$ are analytic at $z = α, α^b, α^{b²},...$ Then the $θ_i = f_i(α)$ are algebraically independent numbers. This was essentially proved by Yu. V. Nesterenko for particular system (*). He gave an ineffective measure of algebraic independence. The purpose of this thesis is to determine an effective measure of algebraic independence for the general case. In certain cases the estimate obtained implies that $(θ₁,...,θ_m)$ has finite transcendence type in the sense of S. Lang.

CONTENTSAcknowledgements...................................................................4I. Introduction§ 1.1. Algebraic independence.................................................5§ 1.2. Notation and some estimates..........................................9II. Formal series§ 2.1. A class to which the solution belong..............................11III. Zero estimates.§ 3.1. The general case ........................................................15§ 3.2. Resultants.....................................................................23§ 3.3. The upper triangular case ...........................................26IV. Preliminaries§ 4.1. Ideals............................................................................30§ 4.2. Some lemmas ..............................................................34V. The main results§ 5.1. Hypothesis Hyp(f,α)......................................................41§ 5.2. Conclusions .................................................................45Appendix.................................................................................53References.............................................................................60

1985 Mathematics Subject Classification 11J85

Détail

Zbl 0707.11050

EUDML-ID : urn:eudml:doc:268434