The Jacobi-Perron Algorithm Its Theory and Application
- PDF / 5,030,343 Bytes
- 165 Pages / 504 x 720 pts Page_size
- 5 Downloads / 191 Views
207 Leon Bernstein Illinois Institute of Technology, Ch icago, ILIUSA
The Jacobi-Perron Algorithm Its Theory and Application
Springer-Verlag Berlin' Heidelberg· New York 1971
AMS Subject Classifications (1970): 12A 99
ISBN 3-540-D5497-9 Springer-Verlag Berlin' Heidelberg· New York ISBN 0-387 -D5497 -9 Springer-Verlag New York . Heidelberg . Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin' Heidelberg 1971. Library of Congress Catalog Card Number70-164956. Printed in Germany. Offsetdruck: Julius Beltz, HemsbachfBergstr.
TABLE OF CONTENTS Introduction • •
..···
Chapter 1 • BASIC CONCEPTS AND RELATIONS § 1 • Definition of the JACOBI-PERRON Algorithm § 2. The Matrices of the JPA § 3. Basic Relations
11 11 12 15
Chapter 2. CONVERGENCE OF JPA • . • . • • . . . • . • • . • § 1. An Analogy with Continued Fractions • • • • § 2. The First Main Convergence Criterion of JPA
19 19 21
C:hapter 3. PERIODICITY OF JPA • . • • • . • • § 1. Two Definitions of Periodicity § 2. The Function f (a(k)) = [a(k~
30 30 32
······
.. ·
····
·......
Chapter 4. SOME SPECIAL CASES OF JPA • • . . . § 1. Periodicity of a
(0)
. d·lCl. t y § 2 • P erlO
0
§ 3. The cases ()(.
='0' D3
f
a
(0)
Eo
e;
E
n-
l'
En-1'
+ 3D
=
g(
o£
and
= 0(
1
(Dn + d) n
49
1
(Dn - iii d)n . . .
=\(j D3
49
56
+ 6D'
68
72 ....·..·· .· 72 79 ····· 88 · · · · 95 ···· UNITS IN ALGEBRAIC NUMBER FIELDS · · · · · · · · 102 102 § 1 • The Characteristic Equation of a Periodic JPA · § 2. Units and Periodicity of the JPA · · · · · · 107 111 § 3. Explicit Units of Algebraic Number Fields
Chapter 5. VARIOUS T-FUNCTIONS § 1 • The Inner T-Function § 2. Irreducibility of Polynomials § 3. Application of the Inner T-Function § 4. The Inner-Outer T-Functions Chapter 6.
§ 4. Sets of Independent Units in Q(w) § 5. Units NOT from the Period of JPA
·
·
117 126
IV Chapter 7. DIOPHANTINE EQUATIONS • . . • • . § 1. Explicit Summation Formulas § 2. Diophantine Equations References •
•• 134 • 134 • • • 143 • 157
Introduction
"Peut-~tre parviendra-t-on ~ deduire de l~ (namely - des circumstances remarquables, aux-quelles donne lieu la reduction des formes, dont les coefficiens dependent des racines d'equations algebraiques ~ coefficiens entiers) un systeme complet de caract~res pour chaque
,
espece de ce genre de
.
,.
quant~tes,
'-
analogue p •. ex. a ceux, que donne la
theorie des fractions continues pour Ie racines des equations du second degre" - with these words Charles Hermite, in one of his number-theoric letters
[14J
to C. G. J. Jacobi, challenged a great
master of ninet
Data Loading...
