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We obtain a theorem concerning the algebraic dependence of the p+1 numbersθ,θ 1 ...θ p subject to the condition that the numbersθ 1 , ...,θ p are algebraically independent and possess a “sufficiently good” estimate of measure of algebraic independence.

We prove general results concerning the algebraic independence of three values of the exponential function. Forβ algebraic and of degree 7 andα algebraic and ≠ 0, 1 there exist among the numbers αβ,...,\(\alpha ^{\beta ^6 } \) three which are algebraically independent. The proof employs a method due to A. O. Gel'fond and N. I. Fel'dman.

A study is made of an approximation by means of algebraic numbers of fixed degree of the roots of the equation P(z, az)=0, a ∉ A, a ≠ 0, 1, where P(x, y) is a polynomial with integral rational coefficients.

The author studies the approximation of fixed-degree transcendental numbers e α a β by algebraic numbers whereα ≠ 0, a ≠ 0; and 1 and irrationalβ are algebraic numbers.

We obtain results showing that transcendental numbers of the form αβ, wherea≠0, 1, β is irrational, anda and β are algebraic numbers, cannot be expressed algebraically in terms of two of the numbers. The proof is carried out by A. O. Gel'fond's method.

Given the three numbers of,a β 1 β 2 , and Ina 2/Ina 1, wherea 1 anda 2 are algebraic numbers whose logarithms are linearly independent in a rational field andΒ is a quadratic irrationality, it is shown that they are not all expressible algebraically in terms of one of them.

Let Γ be an algebraic curve determined over a finite field k = [q]; e,x are subsidiary additive and multiplicative characters of the field k;ϕ,ψ are functions in Γ determined over k and satisfying some natural conditions. If P passes through the points of curve Γ, rational over k, then where constant C depends only on the powers of Γ,ϕ,ψ.

In this paper we prove an “explicit formula” relating the scalar product of the Hecke series of two fields of algebraic numbers to the zeta-function composites of these fields.

We show that the numbers eαaβ are transcendental, where a ≠ 0; 1, is irrational, α≠0, andβ is algebraic.

Suppose Xisametric spacewithmetric d, ~ M cX, andforxEX, d(x, M) = inf[d(x, y)i y~M}, PMz={y~M: d(x, y)= d(x, M)}. The set M is called an existence set [i] (resp., approximatively compact [2]) if for each x~X we have PM x ~ (resp., if for each x~X any sequence "{yn}n~_-,~M such that d(x, yn)-+d(x,M) contains a subsequence converging to an element of M). Every approximatively compact set is an existence set. A special case of an approximatively compact set is a delta-compact set. Let V(z, r) = {y~X: d(z, y)~.~r} (zEX, r~0) A set M is called delta-compact (6-compact, spherically compact) if for each x~ X~M there exists 8>0 such that V(z, d~, M)+~) ~ M is compact. Delta-compact sets were studied in [3-9].