Discriminant equations in diophantine number theory
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Cambridge University Press
2017
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| 245 | 1 | 0 | |a Discriminant equations in diophantine number theory |c Jan-Hendrik Evertse, Leiden University, The Netherlands, Kálmán Győry, University of Debrecen, Hungary |
| 264 | 1 | |a Cambridge |b Cambridge University Press |c 2017 | |
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Datensatz im Suchindex
| _version_ | 1819677840591814656 |
|---|---|
| adam_text | Contents
Preface page xi
Acknowledgments xii
Summary xiii
PART ONE PRELIMINARIES
1 Finite Etale Algebras over Fields 3
1.1 Terminology for Rings and Algebras 3
1.2 Finite Field Extensions 4
1.3 Basic Facts on Finite Etale Algebras over Fields 6
1A Resultants and Discriminants of Polynomials 9
1.5 Characteristic Polynomial, Trace, Norm, Discriminant 11
1.6 Integral Elements and Orders 15
2 Dedekind Domains 17
2.1 Definitions 17
2.2 Ideal Theory of Dedekind Domains 18
2.3 Discrete Valuations 20
2.4 Local ization 21
2.5 Integral Closure in Finite Field Extensions 21
2.6 Extensions of Discrete Valuations 22
2.7 Norms of Ideals 24
2.8 Discriminant and Different 25
2.9 Lattices over Dedekind Domains 27
2.10 Discriminants of Lattices of Etale Algebras 30
3 Algebraic Number Fields 34
3.1 Definitions and Basic Results 34
3.1.1 Absolute Norm of an Ideal 34
v
35
36
37
39
41
44
48
50
52
56
58
60
60
61
64
66
73
74
76
78
80
80
83
85
86
87
89
92
94
104
Contents
ЗЛ .2 Discriminant, Class Number, Unit Group and
Regulator
3.1.3 Explicit Estimates
3.2 Absolute Values: Generalities
3.3 Absolute Values and Places on Number Fields
3.4 5-integers, 5-units and S-norm
3.5 Heights and Houses
3.6 Estimates for Units and 5-units
3.7 Effective Computations in Number Fields and Étale
Algebras
3.7.1 Algebraic Number Fields
3.7.2 Relative Extensions and Finite Étale Algebras
Tools from the Theory of Unit Equations
4.1 Effective Results over Number Fields
4.1.1 Equations in Units of Rings of Integers
4.1.2 Equations with Unknowns from a Finitely
Generated Multiplicative Group
4.2 Effective Results over Finitely Generated Domains
4.3 Ineffective Results, Bounds for the Number of Solutions
PART TWO MONIC POLYNOMIALS AND INTEGRAL
ELEMENTS OF GIVEN DISCRIMINANT, MONOGENIC
ORDERS
Basic Finiteness Theorems
5.1 Basic Facts on Finitely Generated Domains
5.2 Discriminant Forms and Index Forms
5.3 Monogenic Orders, Power Bases, Indices
5.4 Finiteness Results
5.4.1 Discriminant Equations for Monic Polynomials
5.4.2 Discriminant Equations for Integral Elements in
Étale Algebras
5.4.3 Discriminant Form and Index Form Equations
5.4.4 Consequences for Monogenic Orders
Effective Results over Z
6.1 Discriminant Form and Index Form Equations
6.2 Applications to Integers in a Number Field
6.3 Proofs
6.4 Algebraic Integers of Arbitrary Degree
Contents vii
6.5 Proofs 106
6.6 Monic Polynomials of Given Discriminant 108
6.7 Proofs 109
6.8 Notes 113
6.8.1 Some Related Results 113
6.8.2 Generalizations over Z 114
6.8.3 Other Applications 114
7 Algorithmic Resolution of Discriminant Form and Index
Form Equations 117
7.1 Solving Discriminant Form and Index Form Equations
via Unit Equations, A General Approach 118
7.1.1 Quintic Number Fields 121
7.1.2 Examples 133
7.2 Solving Discriminant Form and Index Form Equations
via Thue Equations 137
7.2.1 Cubic Number Fields 138
7.2.2 Quartic Number Fields 138
7.2.3 Examples 142
7.3 The Solvability of Index Equations in Various Special
Number Fields 145
7.4 Notes 146
8 Effective Results over the 5-integers of a Number Field 148
8.1 Results over Zs 149
8.2 Monic Polynomials with 5-integral Coefficients 152
8.3 Proofs 157
8.4 Integral Elements over Rings of 5-integers 172
8.4.1 Integral Elements in Etale Algebras 172
8.4.2 Integral Elements in Number Fields 178
8.4.3 Algebraic Integers of Given Degree 179
8.5 Proofs 182
8.6 Notes 191
8.6.1 Historical Remarks 191
8.6.2 Generalizations and Analogues 192
8.6.3 The Existence of Relative Power Integral Bases 195
8.6.4 Other Applications 195
9 The Number of Solutions of Discriminant Equations 196
9.1 Results over Z 197
9.2 Results over the 5-integers of a Number Field 200
Contents
viii
93 Proof of Theorem 9.2.1 202
9.4 Proof of Theorem 9.2.2 205
9.5 Three Times Monogenic Orders over Finitely Generated
Domains 209
9.6 Notes 218
10 Effective Results over Finitely Generated Domains 222
10.1 Statements of the Results 223
10.1.1 Results for General Domains 224
10.1.2 A Special Class of Integral Domains 226
10.2 The Main Proposition 228
10.3 Rank Estimates for Unit Groups 229
10.4 Proofs of Theorems 10.1.1 and 10.1.2 231
10.5 Proofs of Theorem 10.1.3 and Corollary 10.1.4 236
10.6 Proofs of the Results from Subsection 10.1.2 239
10.7 Supplement: Effective Computations in Finitely
Generated Domains 245
10.7.1 Finitely Generated Fields over Q 245
10.7.2 Finitely Generated Domains over Z 249
10.8 Notes 255
11 Further Applications 257
11.1 Number Systems and Power Integral Bases 257
11.1.1 Canonical Number Systems in Algebraic Number
Fields 258
11.1.2 Proofs 259
11.1.3 Notes 266
11.2 The Number of Generators of an G^-order 268
11.2.1 Notes 271
PART THREE BINARY FORMS OF GIVEN DISCRIMINANT
12 A Brief Overview of the Basic Finiteness Theorems 275
13 Reduction Theory of Binary Forms 278
13.1 Reduction of Binary Forms over Z 279
13.2 Geometry of Numbers over the S-integers 284
13.3 Estimates for Polynomials 290
13.4 Reduction of Binary Forms over the S-integers 293
14 Effective Results for Binary Forms of Given Discriminant 302
14.1 Results over Z 303
Contents
ix
14.2 Results over the S-integers of a Number Field 305
14.3 Applications 307
14.4 Proofs of the Results from Section 14.2 311
Î4.5 Proofs of the Results from Section 14.3 323
14.6 Bounding the Degree of Binary Forms over Z of Given
Discriminant 327
14.7 A Conséquence for Monic Polynomials 330
14.8 Relation between Binary Forms of Given Discriminant
and Unit Equations in Two Unknowns 332
14.9 Decomposable Forms of Given Semi֊Discriminant 333
14.10 Notes 337
14.10.1 Applications to Classical Diophantine
Equations 337
14.10.2 Other Applications 338
14.10.3 Practical Algorithms 338
15 Semi-effective Results for Binary Forms of Given
Discriminant 339
15.1 Results 340
15.2 The Basic Proposition 342
15.3 Construction of the Tuple 343
15.4 Proof of the Basic Proposition 346
15.5 Notes 356
16 Invariant Orders of Binary Forms 358
16.1 Algebras Associated with a Binary Form 359
16.2 Définition of the Invariant Order 361
16.3 Binary Cubic Forms and Cubic Orders 369
17 On the Number of Equivalence Classes of Binary Forms of
Given Discriminant 371
17.1 Results over Z 372
17.2 Results over the S-integers of a Number Field 374
17.3 £2-forms 376
17.4 Local-to-Global Results 378
17.5 Lower Bounds 384
17.6 Counting Equivalence Classes over Discrète Valuation
Domains 386
17.7 Counting Equivalence Classes over Number Fields 395
17.8 Proofs of the Theorems 401
17.9 Finiteness Results over Finitely Generated Domains 403
17.10 Notes 408
X
Contents
18 Further Applications 409
18.1 Root Separation of Polynomials 409
18.1.1 Results for Polynomials over Z 410
18.1.2 Results over Number Fields 41 1
18.1.3 Proof of Theorem 18.1.5 413
18.1.4 Proof of Theorems 18. i .6 and 18.1.7 421
18.1.5 Notes 424
18.2 An Effective Proof of Shafarevich’s Conjecture for
Hyperelliptic Curves 425
18.2.1 Definitions 426
18.2.2 Results 427
18.2.3 Preliminaries 429
18.2.4 Proofs 430
18.2.5 Notes 435
Glossary of Frequently Used Notation 436
References 440
Index 454
Diophantine number theory is an active area that has
seen tremendous progress over the past century. An
important role in this theory is played by discriminant
equations, a class of Diophantine equations with
close ties to algebraic number theory, Diophantine
approximation and Diophantine geometry. Discriminant
equations are about univariate polynomials or binary
forms of given discriminant, considered over various
types of integral domains.
This book is the first comprehensive account of
discriminant equations and their applications. It brings
together many aspects, including effective results over
number fields, effective results over finitely generated
domains, practical algorithms for solving concrete
equations, estimates on the number of solutions,
applications to algebraic integers of given discriminant,
power integral bases, canonical number systems,
algorithms for finding a minimal set of generators of an
order of a number field, root separation of polynomials
and reduction of hyperelliptic curves. The authors’
previous title, Unit Equations in Diophantine Number Theory,
laid the groundwork by presenting important results
that are used as tools in the present book. This material
is briefly summarized in the introductory chapters along
with the necessary basic algebra and algebraic number
theory, making the book accessible to experts and young
researchers alike.
x
x
The Neiu Mathematical Monographs are
dedicated to books containing an
in-depth discussion of a substantial
area of mathematics. They bring the
reader to the forefront of research
by presenting a synthesis of the key
results, while also acknowledging the
wider mathematical context. As well
as being detailed, they are readable
and contain the motivational material
necessary for those entering a field.
For established researchers they are
a valuable resource. Books are edited
and typeset to a high standard and
published in hardback.
Jan-Hendrik Evertse is a number theorist,
working at the Mathematical Institute of Leiden
University. He has written several influential papers
in Diophantine number theory.
Kálmán Győry is Professor Emeritus at the
University of Debrecen, a member of the Hungarian
Academy of Sciences and a well-known researcher
in Diophantine number theory.
Cambridge
UNIVERSITY PRESS
www.cambridge.org
ISBN 978-1-107-09761-2
|
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| id | DE-604.BV044019481 |
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| indexdate | 2024-12-24T05:40:14Z |
| institution | BVB |
| isbn | 9781107097612 |
| language | English |
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| publisher | Cambridge University Press |
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| series | New mathematical monographs |
| series2 | New mathematical monographs |
| spellingShingle | Evertse, Jan Hendrik 1958- Győry, Kálmán 1940- Discriminant equations in diophantine number theory New mathematical monographs Diophantische Gleichung (DE-588)4012386-8 gnd Zahlentheorie (DE-588)4067277-3 gnd |
| subject_GND | (DE-588)4012386-8 (DE-588)4067277-3 |
| title | Discriminant equations in diophantine number theory |
| title_auth | Discriminant equations in diophantine number theory |
| title_exact_search | Discriminant equations in diophantine number theory |
| title_full | Discriminant equations in diophantine number theory Jan-Hendrik Evertse, Leiden University, The Netherlands, Kálmán Győry, University of Debrecen, Hungary |
| title_fullStr | Discriminant equations in diophantine number theory Jan-Hendrik Evertse, Leiden University, The Netherlands, Kálmán Győry, University of Debrecen, Hungary |
| title_full_unstemmed | Discriminant equations in diophantine number theory Jan-Hendrik Evertse, Leiden University, The Netherlands, Kálmán Győry, University of Debrecen, Hungary |
| title_short | Discriminant equations in diophantine number theory |
| title_sort | discriminant equations in diophantine number theory |
| topic | Diophantische Gleichung (DE-588)4012386-8 gnd Zahlentheorie (DE-588)4067277-3 gnd |
| topic_facet | Diophantische Gleichung Zahlentheorie |
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