Оценки многочленов от значений Е-функций

Abstract

The Siegel-Shidlovskii method is one of the main methods in the theory of transcendental numbers. By using this method, it is possible to establish the transcendency and algebraic independence of the values of entire functions belonging to a certain class (so-called E-functions) provided that these functions are algebraically independent over C(z). There are many examples of E-functions used in mathematics: exp(z), sinz, cosz, sinhz, coshz, Bessel functions, Kummer functions, incomplete gamma-function, generalized hypergeometric functions, and some other special functions. By using the Siegel-Shidlovskii method, it is possible to obtain the lower estimates for the moduli of polynomials from the values of E-functions. These estimates are called algebraic independence measures of the numbers. They serve as the quantitative characteristics of algebraic independence. The problem of obtaining the estimates of algebraic independence measures has long been dealt with by many researchers. The first estimates of measures for the values of exp(z) were obtained by E. Borel and K. Mahler, and for the values of Bessel functions such estimates were obtained by C. Siegel. Estimates of algebraic independence measures for the values of general E-functions were established by S. Lang, A.I. Galochkin, A.B. Shidlovskii, and Yu.V. Nesterenko. The constant appearing in the estimate of a measure is called effective if it can be expressed in terms of the parameters characterizing the considered functions and points at which their values are taken. An estimate of a measure is considered to be effective if it contains only effective constants. Generally speaking, the algebraic independence condition of functions is not sufficient for obtaining effective estimates of algebraic independence measures. Some additional conditions must be satisfied for this. The case in which the main totality of functions is algebraically dependent over C(z) is the most complicated one. In a number of the author’s works, the estimates of algebraic independence measures of the values of E-functions were improved. After 1995, no papers have been published on this topic, although it still remains of issue.