On estimates for polynomials in values of $E$-functions

A. B. Shidlovskii

Sbornik. Mathematics, Tome 43 (1982) no. 1, pp. 1-32

Résumé

In this article the notions of measures of linear independence, transcendence and relative transcendence of numbers are generalized to the case when the linear forms and polynomials in the definition have algebraic coefficients. An axiomatization is given for a method of estimating the measures from the values at algebraic points of a set of $E$-functions satisfying linear differential equations with coefficients in the field of rational functions. Several theorems which estimate such measures are proved in the case when the coefficients of the power series of the $E$-functions under consideration, the coefficients of the polynomials in the measures and the values of the argument belong to an arbitrary algebraic number field. Here most of the theorems proved relate to the case when one estimates the measures of a subset of values of $E$-functions, and the basic set of $E$-functions being considered is algebraically dependent over the field of rational functions. As corollaries of these estimates, some important arithmetic properties of the values of sets of functions which are products of powers of these functions are obtained. Bibliography: 31 titles.