Kumiko Nishioka

Arithmetical properties of a certain power series

Abstract The function f(θ, φ; x, y) = Σk = 1∞ Σ1 ≤ m ≤ kθ + φ xkym, where θ > 0 is irrational and φ is real, satisfies Mahler-type functional equations which enable us to represent it by a gap-like series and then by a continued fraction. Using these representations, we describe the sequence {[(k + 1) θ + φ] − [kθ + φ]}k = 1∞ by a chain of substitutions and give algebraic independence results for the values of f(θ, φ, x, y) at some algebraic points when the partial quotients of the continued fraction of θ are unbounded, and irrationality measures for the values at some rational points.

New approach in Mahler's method.

One of the most important motivations to write this paper originates the paper [3] by Loxton and van der Poorten in which they study algebraic independence of the values of Mahler functions. The first sentence in the proof of Lemma 5 in [3] is not clear. In this paper we will prove a theorem which includes the theorem for functions of one variable in [3] by Nesterenkos method in [6]. In [6] Nesterenko found an algebraic independence measure of the values of Mahler functions whose algebraic independence has been already known. Here we shall show that his method is applicable even to the algebraic independence of the values of Mahler functions of more general type.

Algebraic independence of certain power series of algebraic numbers

Abstract Let f(z) = Σk = 0∞ zk!. Then in p-adic field we prove that for any algebraic numbers α1 ,…, αn with 0 α i α j is not a root of unity for i ≠ j. In the complex field we prove the above result only when n = 2, making use of the p-adic field.

P-adic transcendental numbers

Explicit sets of cardinality 2o of p-adic numbers which are algebraically independent over Qp are constructed. Let Qp be the p-adic completion of Q for a prime p. Let Qp be the algebraic closure of Qp, and Cp be its p-adic completion which is an algebraically closed field of cardinality 2O . Let Qunram be the maximum unramiQp fied extension field of Qp. Then Qpunra = QP (W), where W is the set of all roots of unity whose orders are prime to p. Let Cunram be the p-adic closure p of Qunram in C,. Koblitz [1] asked whether Cunram has uncountably infinite transcendence degree over Qp and Cp has uncountably infinite transcendence degree over Cunram. Lampert [2] answered that the transcendence degree of p Cunram over Qp is 2tO and the transcendence degree of Cp over C unram is 2No by constructing sets of algebraically independent numbers of cardinality 2NO. Here we will give more explicit examples of such sets which cannot be obtained by the method in [2]. Theorem. Let K be an intermediate field between Qp and C P. Let ca 1 C.am be in Cp and a,, ... 5 aXnI be algebraically independent over K. Suppose that for i = 1, ... ,m 1 there exist sequences {flik}k>1 in Cp converging to a and a sequence {Sk}k>i of finite subsets of Aut(CP/K({1,lk}l<I<,_l )) which satisfies (1) lim S,jI = oo and ac i a forany , cSk with a f , k--oo (2) max la fliklp =0 (min l<7 lp as k oo where we define the left-hand side of (2) to be 0 if m = 1. Then (xl, .. m are algebraically independent over K. To prove the theorem we need the following lemma which is proved in Koblitz [1]. Received by the editors March 13, 1989. 1980 Matheinatics Subject Classification (1985 Revision). Primary 1 1J6 1. ? 1990 American Mathematical Society 0002-9939/90

Evertse theorem in algebraic independence

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On a problem of mahler for transcendency of function values

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