Methods of the theory of transcendental numbers, Diophantine approximations and solutions of Diophantine equations

1984 ◽  
pp. 217-263
Author(s):  
Gregory Chudnovsky
Keyword(s):  
Diophantine Equations ◽  
Transcendental Numbers ◽  
Diophantine Approximations

Herbrand-Analysen zweier Beweise des Satzes von Roth: Polynomiale Anzahlschranken

1989 ◽  
Vol 54 (1) ◽  
pp. 234-263 ◽  
Author(s):  
H. Luckhardt
Keyword(s):  
Continued Fraction ◽  
Diophantine Equations ◽  
Growth Conditions ◽  
Number Of Solutions ◽  
Exponential Bound ◽  
Low Degree ◽  
Diophantine Approximations ◽  
Finiteness Theorems ◽  
Fourth Root ◽  
Simple Growth

AbstractA previously unexplored method, combining logical and mathematical elements, is shown to yield substantial numerical improvements in the area of Diophantine approximations. Kreisel illustrated the method abstractly by noting that effective bounds on the number of elements are ensured if Herbrand terms from ineffective proofs ofΣ2-finiteness theorems satisfy certain simple growth conditions. Here several efficient growth conditions for the same purpose are presented that are actually satisfied in practice, in particular, by the proofs of Roth's theorem due to Roth himself and to Esnault and Viehweg. The analysis of the former yields an exponential bound of order exp(70ε−2d2) in place of exp(285ε−2d2) given by Davenport and Roth in 1955, whereαis (real) algebraic of degreed≥ 2 and ∣α−pq−1∣ <q−2−ε. (Thus the new bound is less than the fourth root of the old one.) The new bounds extracted from the other proof arepolynomial of low degree(inε−1and logd). Corollaries: Apart from a new bound for the number of solutions of the corresponding Diophantine equations and inequalities (among them Thue's inequality), log logqν, <Cα, εν5/6+ε, whereqνare the denominators of the convergents to the continued fraction ofα.

An application of the Thue–Siegel–Roth theorem to elliptic functions

1971 ◽  
Vol 69 (1) ◽  
pp. 157-161 ◽  
Author(s):  
J. Coates
Keyword(s):  
Number Theory ◽  
Diophantine Equations ◽  
Elliptic Functions ◽  
Rational Numbers ◽  
Algebraic Numbers ◽  
Number Of Solutions ◽  
Transcendental Numbers ◽  
Linearly Independent ◽  
Image Position ◽  
Effective Proof

Let α1, …, αn be n ≥ 2 algebraic numbers such that log α1,…, log αn and 2πi are linearly independent over the field of rational numbers Q. It is well known (see (6), Ch. 1) that the Thue–Siegel–Roth theorem implies that, for each positive number δ, there are only finitely many integers b1,…, bn satisfyingwhere H denotes the maximum of the absolute values of b1, …, bn. However, such an argument cannot provide an explicit upper bound for the solutions of (1), because of the non-effective nature of the theorem of Thue–Siegel–Roth. An effective proof that (1) has only a finite number of solutions was given by Gelfond (6) in the case n = 2, and by Baker(1) for arbitrary n. The work of both these authors is based on arguments from the theory of transcendental numbers. Baker's effective proof of (1) has important applications to other problems in number theory; in particular, it provides an algorithm for solving a wide class of diophantine equations in two variables (2).

scholarly journals Mathematical Reminiscences: How to Keep the Pot Boiling

2013 ◽  
Vol Volume 34-35 ◽  
Author(s):  
K Ramachandra
Keyword(s):  
Number Theory ◽  
Zeta Function ◽  
Riemann Zeta Function ◽  
Diophantine Equations ◽  
Analytic Number Theory ◽  
Transcendental Numbers ◽  
Analytic Number ◽  
The Riemann Zeta Function ◽  
International Audience ◽  
Riemann Zeta

International audience Analytic number theory deals with the application of analysis, both real and complex, to the study of numbers. It includes primes, transcendental numbers, diophantine equations and other questions. The study of the Riemann zeta-function $\zeta(s)$ is intimately connected with that of primes. \par In this note, edited specially for this volume by K. Srinivas, some problems from a handwritten manuscript of Ramachandra are listed.

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