Riemann-Roch Spaces and Computation

by Paraskevas Alvanos

Overview

The book focuses on the educational perspective of Riemann-Roch spaces and the computation of algebraic structures connected to the Riemann-Roch theorem, which is a useful tool in fields of complex analysis and algebraic geometry. On one hand, the theorem connects the Riemann surface with its topological genus, and on the other it allows us to compute the algebraic function field spaces. In the first part of this book, algebraic structures and some of their properties are presented. The second part shows efficient algorithms and examples connected to Riemann-Roch spaces. What is important, a variety of examples with codes of algorithms are given in the book, covering the majority of the cases.

Product Details

ISBN-13:	9783110439489
Publisher:	De Gruyter Poland
Publication date:	07/25/2016
Sold by:	Barnes & Noble
Format:	eBook
Pages:	151
File size:	6 MB
Age Range:	18 Years

About the Author

Paraskevas Alvanos, Model Experimental High School of Thessaloniki, Greece

Table of Contents

Preface viii

Part I Riemann-Roch Spaces

1 Elements of Algebra 2

1.1 Domains 2

1.2 Rings and Fields 9

1.3 Norm and Trace 13

1.4 Group of Units 15

2 Function Fields and Curves 17

2.1 Algebraic Function Fields 17

2.2 Affine Curves 19

2.3 Projective Curves 23

2.4 Genus of a Curve 27

3 Riemann-Roch Spaces 32

3.1 Valuation Rings 32

3.2 Places and Divisors 36

3.3 Riemann-Roch Space 39

3.4 Holomorphy Rings 41

4 Integral Domains 44

4.1 Fractional Ideals 44

4.2 Localization 46

4.3 R-module V 50

Part II Computation

5 Computing Integral Bases 54

5.1 Discriminant 54

5.2 Ideal Basis 56

5.3 Idealizer 58

5.4 Radical 60

5.5 Computing the Discriminants Ideal Radical 61

5.6 Computing Idealizers and Inverse Ideals 64

5.7 Trager's Algorithm 66

5.8 Examples 67

6 Computing Riemann-Roch Spaces 73

6.1 Reduced Basis Algorithms 73

6.2 The Dimension of L(D) 79

6.3 Computing L(D) 81

7 Computing Resultant and Norm Form Equations 85

7.1 Resultant 85

7.2 Computing N_L/K(X₁a₁+…+X_na_n) = K 90

7.3 Examples 92

8 Computing Integral Points on Rational Curves 98

8.1 Integral Points on Curves 98

8.2 Integral Points on Curves with |σ_∞(C)| ≥ 3 99

8.3 Integral Points on Curves with |σ_∞(C)| = 2 (i) 106

8.4 Integral Points on Curves with |σ_∞(C)| = 2 (ii) 109

8.5 Integral Points on Curves with |σ_∞(C)| = 1 112

Appendices 115

A Unimodular Matrices 115

A.1 Transformation Matrix 115

A.2 Unimodular Transformations 116

B Algorithm's Codes 118

B.1 Integral Points 3 Algorithm's Code 118

B.2 Integral Points 2i Algorithm's Code 126

B.3 Integral Points 2ii Algorithm's Code 129

B.4 Integral Points 1 Algorithm's Code 133

Bibliography 136

Index 139

Index of Algorithms 141