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

Abstract

This thesis is concerned with the problem of determining a measure of algebraic independence for a particular m-tuple $\theta\sb1, \... \theta\sb{\rm m}$ of complex numbers. Specifically let K be a number field and let f$\sb1$(z),... f$\sb{\rm m}$(z) be elements of K((z)) algebraically independent over K(z) satisfying equations of the form.(UNFORMATTED TABLE OR EQUATION FOLLOWS)$$\eqalign{\rm f\sb{j}(z\sp{b})&=\rm\sum\limits\sbsp{i=1}{m}\ f\sb{i}(z)a\sb{ij}(z)+b\sb{j}(z)\cr&\qquad\qquad\qquad\rm (j = 1,\... m)(\*)\cr}$$(TABLE/EQUATION ENDS)for b $\geq$ 2, a$\sb{\rm ij}$(z), b$\sb{\rm j}$(z) in K(z). Suppose finally that $\alpha\in\kappa$ is such that 0 $<$ $\vert\alpha\vert$ $<$ 1, the f$\sb{\rm j}$(z) converge at z = $\alpha$ and the a$\sb{\rm ij}$(z), b$\sb{\rm j}$(z) are analytic at z = $\alpha, \alpha\sp{\rm b}, \alpha\sp{\rm b\sp2},\....$ Then the $\theta$ = f$\sb{\rm i}(\alpha$) are algebraically independent numbers. This was essentially proved by Yu V. Nesterenko for particular systems (*). 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 ($\theta\sb1 \... \theta\sb{\rm m}$) has finite transcendence type in the sense of S. Lang.