Diophantine approximation on linear algebraic groups: transcendence properties of the exponential function in several variables
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Berlin [u.a.]
Springer
2000
|
| Schriftenreihe: | Grundlehren der mathematischen Wissenschaften
326 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XXIII, 633 S. |
| ISBN: | 3540667857 |
Internformat
MARC
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| 245 | 1 | 0 | |a Diophantine approximation on linear algebraic groups |b transcendence properties of the exponential function in several variables |c Michel Waldschmidt |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2000 | |
| 300 | |a XXIII, 633 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Grundlehren der mathematischen Wissenschaften |v 326 | |
| 650 | 4 | |a Approximation diophantienne | |
| 650 | 7 | |a Approximation diophantienne |2 ram | |
| 650 | 7 | |a Approximation diophantienne |2 rasuqam | |
| 650 | 7 | |a Diofantische benadering |2 gtt | |
| 650 | 7 | |a Groupe linéaire algébrique |2 rasuqam | |
| 650 | 7 | |a Groupes algébriques linéaires |2 ram | |
| 650 | 4 | |a Groupes linéaires algébriques | |
| 650 | 7 | |a Lineaire algebra |2 gtt | |
| 650 | 4 | |a Diophantine approximation | |
| 650 | 4 | |a Linear algebraic groups | |
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Datensatz im Suchindex
| _version_ | 1804127842159558656 |
|---|---|
| adam_text | Table of Contents
Prerequisites XIII
Notation XIX
1. Introduction and Historical Survey 1
1.1 Liouville, Hermite, Lindemann, Gel fond, Baker 1
1.2 Lower Bounds for |af ••• a* 11 6
1.3 The Six Exponentials Theorem and the Four Exponentials
Conjecture 13
1.4 Algebraic Independence of Logarithms 15
1.5 Diophantine Approximation on Linear Algebraic Groups 19
Exercises 21
Parti. Transcendence
2. Transcendence Proofs in One Variable 29
2.1 Introduction to Transcendence Proofs 29
2.2 Auxiliary Lemmas 33
2.3 Schneider s Method with Alternants Real Case 37
2.4 Gel fond s Method with Interpolation Determinants Real Case .. 43
2.5 Gel fond Schneider s Theorem in the Complex Case 49
2.6 Hermite Lindemann s Theorem in the Complex Case 55
Exercises 59
3. Heights of Algebraic Numbers 65
3.1 Absolute Values on a Number Field 66
3.2 The Absolute Logarithmic Height (Weil) 75
3.3 Mahler s Measure 78
3.4 Usual Height and Size 80
3.5 Liouville s Inequalities 82
3.6 Lower Bound for the Height 86
Open Problems 105
Exercises 106
X Table of Contents
Appendix Inequalities Between Different Heights of a Polynomial
From a Manuscript by Alain Durand 113
4. The Criterion of Schneider Lang 115
4.1 Algebraic Values of Entire Functions Satisfying Differential
Equations 115
4.2 First Proof of Baker s Theorem 118
4.3 Schwarz Lemma for Cartesian Products 122
4.4 Exponential Polynomials 130
4.5 Construction of an Auxiliary Function 131
4.6 DirectProof of Corollary 4.2 136
Exercises 141
Part II. Linear Independence of Logarithms and Measures
5. Zero Estimate, by Damien Roy 147
5.1 The Main Result 147
5.2 Some Algebraic Geometry 150
5.3 The Group G and its Algebraic Subgroups 156
5.4 Proof of the Main Result 164
Exercises 166
6. Linear Independence of Logarithms of Algebraic Numbers 169
6.1 Applying the Zero Estimate 170
6.2 Upper Bounds for Alternants in Several Variables . 175
6.3 A Second Proof of Baker s Homogeneous Theorem 181
Exercises 184
7. Homogeneous Measures of Linear Independence 187
7.1 Statement of the Measure 187
7.2 Lower Bound for a Zero Multiplicity 192
7.3 Upper Bound for the Arithmetic Determinant 195
7.4 Construction of a Nonzero Determinant 199
7.5 The Transcendence Argument — General Case 203
7.6 Proof of Theorem 7.1 — General Case 208
7.7 The Rational Case: Fel dman s Polynomials 214
7.8 Linear Dependence Relations between Logarithms 222
Open Problems 227
Exercises 227
Table of Contents XI
Part ID. Multiplicities in Higher Dimension
8. Multiplicity Estimates, by Damien Roy 231
8.1 The Main Result 231
8.2 Some Commutative Algebra 234
8.3 The Group G and its Invariant Derivations 238
8.4 Proof of the Main Result 245
Exercises 247
9. Refined Measures 251
9.1 Second Proof of Baker s Nonhomogeneous Theorem 252
9.2 Proof of Theorem 9.1 262
9.3 Value of C(m) 286
9.4 Corollaries 302
Exercises 314
10. On Baker s Method 317
10.1 Linear Independence of Logarithms of Algebraic Numbers 317
10.2 Baker s Method with Interpolation Determinants 329
10.3 Baker s Method with Auxiliary Function 356
10.4 The State of the Art 360
Exercises 371
Part IV. The Linear Subgroup Theorem
11. Points Whose Coordinates are Logarithms of Algebraic Numbers .. 375
11.1 Introduction 375
11.2 One Parameter Subgroups 379
11.3 Six Variants of the Main Result 381
11.4 Linear Independence of Logarithms 387
11.5 Complex Toruses 394
11.6 Linear Combinations of Logarithms with Algebraic Coefficients .. 398
11.7 Proof of the Linear Subgroup Theorem 404
Exercises 411
12. Lower Bounds for the Rank of Matrices 417
12.1 Entries are Linear Polynomials 418
12.2 Entries are Logarithms of Algebraic Numbers 432
12.3 Entries are Linear Combinations of Logarithms 435
12.4 Assuming the Conjecture on Algebraic Independence of
Logarithms 437
12.5 Quadratic Relations 438
Exercises 441
XII Table of Contents
Part V. Simultaneous Approximation of Values of the Exponential Function in
Several Variables
13. A Quantitative Version of the Linear Subgroup Theorem 445
13.1 The Main Result 447
13.2 Analytic Estimates 450
13.3 Exponential Polynomials 459
13.4 Proof of Theorem 13.1 464
13.5 Directions for Use 471
13.6 Introducing Feld man s Polynomials 476
13.7 Duality: the Fourier Borel Transform 480
Exercises 490
14. Applications to Diophantine Approximation 495
14.1 A Quantitative Refinement to Gel fond Schneider s Theorem .... 496
14.2 A Quantitative Refinement to Hermite Lindemann s Theorem .... 510
14.3 Simultaneous Approximation in Higher Dimension 521
14.4 Measures of Linear Independence of Logarithms (Again) 536
Open Problems 547
Exercises 549
15. Algebraic Independence 555
15.1 Criteria: Irrationality, Transcendence, Algebraic Independence ... 555
15.2 From Simultaneous Approximation to Algebraic Independence ... 569
15.3 Algebraic Independence Results: Small Transcendence Degree ... 587
15.4 Large Transcendence Degree: Conjecture on Simultaneous
Approximation 594
15.5 Further Results and Conjectures 598
Exercises 606
References 615
Index 629
|
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| institution | BVB |
| isbn | 3540667857 |
| language | English |
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| physical | XXIII, 633 S. |
| publishDate | 2000 |
| publishDateSearch | 2000 |
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| publisher | Springer |
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| series | Grundlehren der mathematischen Wissenschaften |
| series2 | Grundlehren der mathematischen Wissenschaften |
| spelling | Waldschmidt, Michel 1946- Verfasser (DE-588)1022267817 aut Diophantine approximation on linear algebraic groups transcendence properties of the exponential function in several variables Michel Waldschmidt Berlin [u.a.] Springer 2000 XXIII, 633 S. txt rdacontent n rdamedia nc rdacarrier Grundlehren der mathematischen Wissenschaften 326 Approximation diophantienne Approximation diophantienne ram Approximation diophantienne rasuqam Diofantische benadering gtt Groupe linéaire algébrique rasuqam Groupes algébriques linéaires ram Groupes linéaires algébriques Lineaire algebra gtt Diophantine approximation Linear algebraic groups Diophantische Approximation (DE-588)4135760-7 gnd rswk-swf Lineare algebraische Gruppe (DE-588)4295326-1 gnd rswk-swf Lineare algebraische Gruppe (DE-588)4295326-1 s Diophantische Approximation (DE-588)4135760-7 s DE-604 Grundlehren der mathematischen Wissenschaften 326 (DE-604)BV000000395 326 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008954195&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Waldschmidt, Michel 1946- Diophantine approximation on linear algebraic groups transcendence properties of the exponential function in several variables Grundlehren der mathematischen Wissenschaften Approximation diophantienne Approximation diophantienne ram Approximation diophantienne rasuqam Diofantische benadering gtt Groupe linéaire algébrique rasuqam Groupes algébriques linéaires ram Groupes linéaires algébriques Lineaire algebra gtt Diophantine approximation Linear algebraic groups Diophantische Approximation (DE-588)4135760-7 gnd Lineare algebraische Gruppe (DE-588)4295326-1 gnd |
| subject_GND | (DE-588)4135760-7 (DE-588)4295326-1 |
| title | Diophantine approximation on linear algebraic groups transcendence properties of the exponential function in several variables |
| title_auth | Diophantine approximation on linear algebraic groups transcendence properties of the exponential function in several variables |
| title_exact_search | Diophantine approximation on linear algebraic groups transcendence properties of the exponential function in several variables |
| title_full | Diophantine approximation on linear algebraic groups transcendence properties of the exponential function in several variables Michel Waldschmidt |
| title_fullStr | Diophantine approximation on linear algebraic groups transcendence properties of the exponential function in several variables Michel Waldschmidt |
| title_full_unstemmed | Diophantine approximation on linear algebraic groups transcendence properties of the exponential function in several variables Michel Waldschmidt |
| title_short | Diophantine approximation on linear algebraic groups |
| title_sort | diophantine approximation on linear algebraic groups transcendence properties of the exponential function in several variables |
| title_sub | transcendence properties of the exponential function in several variables |
| topic | Approximation diophantienne Approximation diophantienne ram Approximation diophantienne rasuqam Diofantische benadering gtt Groupe linéaire algébrique rasuqam Groupes algébriques linéaires ram Groupes linéaires algébriques Lineaire algebra gtt Diophantine approximation Linear algebraic groups Diophantische Approximation (DE-588)4135760-7 gnd Lineare algebraische Gruppe (DE-588)4295326-1 gnd |
| topic_facet | Approximation diophantienne Diofantische benadering Groupe linéaire algébrique Groupes algébriques linéaires Groupes linéaires algébriques Lineaire algebra Diophantine approximation Linear algebraic groups Diophantische Approximation Lineare algebraische Gruppe |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008954195&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000395 |
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