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## Algebraic independence of the values at algebraic points of a class of functions considered by Mahler

Autorzy
Seria
Rozprawy Matematyczne tom/nr w serii: 303 wydano: 1990
Zawartość
Warianty tytułu
Abstrakty
EN
This thesis is concerned with the problem of determining a measure of algebraic independence for a particular m-tuple θ₁,...,$θ_m$ of complex numbers. Specifically, let K be a number field and let f₁(z),...,$f_m(z)$ be elements of K[[z]] algebraically independent over K(z) satisfying equations of the form
(*) $f_j(z^b) = ∑^m_{i=1} f_i(z)a_{ij}(z) + b_j(z)$ (j = i,...,m)
for b ≥ 2, $a_{ij}(z)$, $b_j(z)$ in K(z). Suppose finally that α ∈ K is such that 0 < |α| < 1, the $f_j(z$) converge at z = α and the $a_{ij}(z)$, $b_j(z)$ are analytic at $z = α, α^b, α^{b²},...$ Then the $θ_i = f_i(α)$ are algebraically independent numbers. This was essentially proved by Yu. V. Nesterenko for particular system (*). He gave an ineffective measure of algebraic independence. The purpose of this thesis is to determine an effective measure of algebraic independence for the general case. In certain cases the estimate obtained implies that $(θ₁,...,θ_m)$ has finite transcendence type in the sense of S. Lang.
EN

CONTENTS
Acknowledgements...................................................................4
I. Introduction
§ 1.1. Algebraic independence.................................................5
§ 1.2. Notation and some estimates..........................................9
II. Formal series
§ 2.1. A class to which the solution belong..............................11
III. Zero estimates.
§ 3.1. The general case ........................................................15
§ 3.2. Resultants.....................................................................23
§ 3.3. The upper triangular case ...........................................26
IV. Preliminaries
§ 4.1. Ideals............................................................................30
§ 4.2. Some lemmas ..............................................................34
V. The main results
§ 5.1. Hypothesis Hyp(f,α)......................................................41
§ 5.2. Conclusions .................................................................45
Appendix.................................................................................53
References.............................................................................60
Słowa kluczowe
Tematy
Kategoryzacja MSC:
Miejsce publikacji
Warszawa
Seria
Rozprawy Matematyczne tom/nr w serii: 303
Liczba stron
61
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCCIII
Daty
wydano
1990
Twórcy
autor
• Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1003, U.S.A.
• 57 Arapito Rd, Titirangi, Auckland 7, New Zealand
Bibliografia
• [В] P.-G. Becker-Landeck, Maße für algebraische Unabhängigkeit nach einer Methode von Mahler, Acta Arith. 50 (1988), 279-293.
• [Br] W. D. Brownawell, Effectively in independence measures for values of E-functions, J. Austral. Math. Soc. Ser. 39 (1985), 227-240.
• [Ch] G. Chudnovsky, Contributions to the Theory of Transcendental Numbers, Surveys and Monographs, 19, AMS, 1984.
• [G] A. I. Galochkin, A transcendence measure for the values of functions satisfying certain functional equations, Mat. Zametki 27 (1980), 175-183; English transl. in Math. Notes 27 (1980).
• [K] К. K. Kubota, On the algebraic independence of holomorphic solutions of certain functional equations and their values, Math. Ann. 227 (1970), 9-50.
• [La1] S. Lang, Introduction to Transcendental Numbers, Addison-Wesley, 1966.
• [La2] S. Lang, A transcndence measure for E-functions, Mathematika 9 (1962), 157-161.
• [L] J. H. Loxton, Automata and Transcendence, Chapter 13 of: New Advances in Transcendence Theory, A. Baker (ed.), Cambridge Univ. Press, 1988.
• [LP1] J. H. Loxton and A. J. van der Poorten, Algebraic independence properties of the Fredholm series, J. Austral. Math. Soc. Ser. A 26 (1978), 31-45.
• [LP2] J. H. Loxton and A. J. van der Poorten, Arithmetic properties of certain functions in several variables III, Bull. Austral. Math. Soc. 16 (1977), 15-47.
• [LP3] J. H. Loxton and A. J. van der Poorten, A class f hypertranscendental functions, Aequationes Math. 16 (1977), 93-106.
• [Ma1] K. Mahler, Remarks on a paper by W. Schwarz, J. Number Theory 1 (1969), 512-521.
• [Ma2] K. Mahler, Arithmetische Eigenschaften der Lösungen einer Klasse von Funktionalgleichungen, Math. Ann. 101 (1929), 342-366.
• [Ma3] K. Mahler, Arithmetische Eigenschaften einer Klasse transzendental-transzendenter Funktionen, Math. Z. 32 (1930), 545-585.
• [Ma4] K. Mahler, Über das Verschwinden von Potenzreihen mehrerer Veränderlichen in speziellen Punktfolgen, Math. Ann. 103 (1930), 573-587.
• [M1] W. Miller, Transcendence measures for values of analytic solutions to certain functional equations, Ph. D. Thesis, University of Michigan 1979.
• [M2] W. Miller, Transcendence measures by a method of Mahler, J. Austral. Math. Soc. Ser. A 32 (1982), 68-72.
• [Ne1] Yu. V. Nesterenko, Estimates for the orders of zeros of functions of a certain class and applications in the theory of transcendental numbers, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), 253-284; English transl. in Math. USSR-Izv. 11 (1977).
• [Ne2] Yu. V. Nesterenko, Estimates for the characteristic function of a prime ideal, Mat. Sb. 123 (165)
• (1984), 11-49; English transl. in Math. USSR-Sb. 51 (1) (1985).
• [Ne3] Yu. V. Nesterenko, On algebraic independence of algebraic powers of algebraic numbers, Mat. Sb. 123 (165) (1984), 435-459; English transl. in Math. USSR-Sb. 51 (1985).
• [Ne4] Yu. V. Nesterenko, On a measure of the algebraic independence of the values of certain functions, Mat. Sb. 128 (170) (1985), 545-568; English transl. in Math. USSR-Sb. 56 (2) (1987).
• [Sh] A. V. Shidlovskiĭ, On criteria for algebraic independence of a class of entire functions, Izv. Akad. Nauk SSSR Ser. Mat. 23 (1959), 35-66; English transl. in Amer. Math. Soc. Transl. (2) 22 (1962).
• [Sie] C. L. Siegel, Approximation algebraischer Zahlen, Math. Z. 10 (1921), 173-213.
• [Wae] B. L. van der Waerden, Modern Algebra, Vols. 1 and 2, F. Ungar, New York 1950 and 1953.
• [Wa] M. Waldschmidt, Nombres Transcendants, Springer, Berlin-New York 1974.
Języki publikacji
 EN
Uwagi
1985 Mathematics Subject Classification 11J85