用逼近方法建立代数无关性(Ⅱ)

设ｆ（ｘ）是以ａｎｘ为元素的连分式,其中系数ａｎ是正代数数．应用［１］中的判别法则,在某些仅与ａｎ有关的条件下证明了对任何绝对值互异的非零实代数数ａ１,…,ａｓ值ｆ（ａ１）代数无关．还对某些实超越数建立了的代数无关性．     

用逼近方法建立代数无关性

应用逼近方法建立了一个关于复数代数无关性的一般性判别法则，并用来研究某些缺项级数在代数点和超越点上值的代数无关性．     

某些超越连分数的代数无关性(Ⅱ)

本文证明了某些以ａｎｘ（ａｎ为正代数数）为元素的连分式在代数点和超越点上值的代数无关性．特别地,某些关于简单连分数的代数无关性结果被扩充到更广泛的情形。     

Algebraic independence by approximation method

By the use of approximation method a general criterion of algebraic independence of complex numbers is established. As its application the algebraic independence of values of certain gap series at algebraic and transcendental points is given. Subject supported by the National Natural Science Foundation of China     

某些无穷乘积的代数无关性(Ⅱ)

本文证明了某些具有代数系数的无穷乘积和幂级数定义的函数在某些代数数和超越数上值的代数无关性.     

代数系数幂级数在超越数上值的代数无关性

本文证明了一类具有代数系数的幂级数在超越数上值约代数无关性．     

Mahler型函数值的代数无关度量

利用初等的结式方法研究满足多项式形式的函数方程组的Mahler型函数的零点估计,给出了满足非线性函数方程组的Mahler型函数在代数点值的代效无关度量．     

Criteria of algebraic independence with multiplicities and interpolation determinants

We generalize Gel'fond's criterion of algebraic independence by taking into account the values of the derivatives of the polynomials, and show how the new criterion applies to proving results of algebraic independence using interpolation determinants. We also establish a new result of approximation of a transcendental number by algebraic numbers of bounded degree and size. It contains an earlier result of E. Wirsing and also a result announced by A. Durand.

     

On a criterion of linear independence of p-adic numbers

General theorems giving sufficient conditions for linear independence of p-adic numbers over algebraic number fields are proved.     

Algebraic independence of exponents

Three theorems are obtained for the algebraic independence of some numbers related to exponential functions. Theorems 1 and 3 are extensions of the well-known Gelfond results.     

Algebraic independence of some values of the exponential function

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 α_β,..., \(\alpha ^{\beta ^6 } \) three which are algebraically independent. The proof employs a method due to A. O. Gel'fond and N. I. Fel'dman.     

Algebraic Independence of Certain Generalized Mahler Type Numbers

In this paper the generalized Mahler type number Mh（g;A,T） is defined, and in the case of multiplicatively dependent parameters gi, hi（1 ≤ i ≤ s） the algebraic independence of the numbers Mhi （gi; A, T）（1 ≤ i ≤ s） is proved, where A and T are certain infinite sequences of non-negative integers and of positive integers, respectively. Furthermore, the algebraic independence result on values of a certain function connected with the generalized Mahler type number and its derivatives at algebraic numbers is also given.     

Über lineare Unabhängigkeit in algebraischen Zahlkörpern

Quantitative and qualitative criteria for linear independence of numbers over an algebraic number field are given, which depend on methods of linear elimination theory. As an application a quantitative version of the Theorem of Lindemann-Weierstrass is proved.     

Algebraic Independence of Reciprocal Sums of Binary Recurrences II

 Algebraic independence of the numbers for various d and l, where is a periodic sequence of algebraic numbers and is a sequence of integers satisfying a binary linear recurrence relation, is studied by Mahler’s method.     

Algebraic independence of power series generated by linearly independent positive numbers

In this paper we establish, using Mahler’s method, the algebraic independence of the values at an algebraic number of power series closely related to decimal expansion of linearly independent positive numbers. First we consider a simpler case in Theorem 1 and then generalize it to Theorem 3, which includes Nishioka’s result quoted as Theorem 2 of this paper. Lemma 7 plays an essential role in the proof of Theorems 1 and 3.     

Algebraische Unabhängigkeit der Werte gewisser Lückenreihen in nicht-archimedisch bewerteten Körpern

In 1844 Liouville proved the transcendence of α = ∑_h≥1 10^{?h h!} over Q. The number α can be considered as the value of the gap power series ∧(x) =∑_h≥1 at tne point 1/10 Since then, the above result has been generalized in this direction by different authors by applying improved “Liouville-estimates”. For instance, in 1973 Cijsouw and Tijdeman [2] showed that a gap series with algebraic coefficients takes on transcendental values (over Q) at non-zero algebraic points under some conditions on the growth of the coefficients and the gaps. In 1988 Bundschuh [1] resp. Zhu [9] proved the algebraic independence (over Q) of the values of several gap series at different algebraic points. In particular this result includes the algebraic independence of A(α₁),…, α(α_s) for non-zero algebraic numbers α₁,…, α_s of distinct absolute values less than 1. Moreover in [1] a set of continuum-many algebraically independent numbers was constructed. In 1978 Geijsel [4] obtained a result analogous to that of Cijsouw and Tijdeman underlying a non-archimedian valued function field over a finite field, and in 1983 Sieburg [7] was concerned with the algebraic independence of “Liouville-series” in non-archimedian valued fields of characteristic zero. In this paper some of the results of [1] resp. [9] will be transfered to the situation of some non-archimedian valued fields. If the characteristic of the field is prime, we have to require stronger conditions as in the “classical case”. An example shows that in this case the numbers A(c*i),..., A(aa) need not to be algebraically independent. But a set of continuum-many algebraically independent numbers still exists. In characteristic zero, results of the same kind will be obtained like in the “classical case”.     

Algebraic independence results for the values of certain Mahler functions and their application to infinite products

In this paper we establish algebraic independence criteria for the values at an algebraic point of Mahler functions each of which satisfies either a multiplicative type of functional equation or an additive one. As application we construct, using a linear recurrence sequence, an entire function defined by an infinite product such that its values as well as its all successive derivatives at algebraic points other than its zeroes are algebraically independent. Zeroes of such an entire function form a subsequence of the linear recurrence sequence. We prove the algebraic independency by reducing those values at algebraic points to those of Mahler functions.     

Algebraic Independence of Reciprocal Sums of Binary Recurrences II

 Algebraic independence of the numbers for various d and l, where is a periodic sequence of algebraic numbers and is a sequence of integers satisfying a binary linear recurrence relation, is studied by Mahler’s method. Received 25 August 2000; in revised form 8 January 2002     

Algebraic Independence by Approximation Method (II)

Let f (x) be a continued fraction with elements a _n x, where coefficients a _n are positive algebraic numbers. Using the criterion of [l] for any nonzero real algebraic numbers α₁,...,α_s with distinct absolute values the algebraic independence of the values f(α₁), ..., f(α_s) is proved under certain assumption concerning only with a _n . For some transcendental numbers ξ the algebraic independence of values f(ξ^j)(j∈ℤ) is also established. Received March 27, 1998, Accepted September 28, 1998     

ON THE DIFFERENTIAL AND DIFFERENCE INDEPENDENCE OF Γ AND ζ

In this paper, we study the algebraic differential and the difference independence between the Riemann zeta function and the Euler gamma function. It is proved that the Riemann zeta function and the Euler gamma function cannot satisfy a class of nontrivial algebraic differential equations and algebraic difference equations.