The Resource Algebraic curves over a finite field, J.W.P. Hirschfeld, G. Korchmaros, F. Torres
Algebraic curves over a finite field, J.W.P. Hirschfeld, G. Korchmaros, F. Torres
 Summary
 This book provides an accessible and selfcontained introduction to the theory of algebraic curves over a finite field, a subject that has been of fundamental importance to mathematics for many years and that has essential applications in areas such as finite geometry, number theory, errorcorrecting codes, and cryptology. Unlike other books, this one emphasizes the algebraic geometry rather than the function field approach to algebraic curves. The authors begin by developing the general theory of curves over any field, highlighting peculiarities occurring for positive characteristi
 Language
 eng
 Extent
 1 online resource (717 pages).
 Contents

 Cover; Title; Copyright; Dedication; Contents; Preface; PART 1. GENERAL THEORY OF CURVES; Chapter 1. Fundamental ideas; 1.1 Basic definitions; 1.2 Polynomials; 1.3 Affine plane curves; 1.4 Projective plane curves; 1.5 The Hessian curve; 1.6 Projective varieties in higherdimensional spaces; 1.7 Exercises; 1.8 Notes; Chapter 2. Elimination theory; 2.1 Elimination of one unknown; 2.2 The discriminant; 2.3 Elimination in a system in two unknowns; 2.4 Exercises; 2.5 Notes; Chapter 3. Singular points and intersections; 3.1 The intersection number of two curves; 3.2 Bézout's Theorem
 3.3 Rational and birational transformations3.4 Quadratic transformations; 3.5 Resolution of singularities; 3.6 Exercises; 3.7 Notes; Chapter 4. Branches and parametrisation; 4.1 Formal power series; 4.2 Branch representations; 4.3 Branches of plane algebraic curves; 4.4 Local quadratic transformations; 4.5 Noether's Theorem; 4.6 Analytic branches; 4.7 Exercises; 4.8 Notes; Chapter 5. The function field of a curve; 5.1 Generic points; 5.2 Rational transformations; 5.3 Places; 5.4 Zeros and poles; 5.5 Separability and inseparability; 5.6 Frobenius rational transformations
 5.7 Derivations and differentials5.8 The genus of a curve; 5.9 Residues of differential forms; 5.10 Higher derivatives in positive characteristic; 5.11 The dual and bidual of a curve; 5.12 Exercises; 5.13 Notes; Chapter 6. Linear series and the RiemannRoch Theorem; 6.1 Divisors and linear series; 6.2 Linear systems of curves; 6.3 Special and nonspecial linear series; 6.4 Reformulation of the RiemannRoch Theorem; 6.5 Some consequences of the RiemannRoch Theorem; 6.6 The Weierstrass Gap Theorem; 6.7 The structure of the divisor class group; 6.8 Exercises; 6.9 Notes
 Chapter 7. Algebraic curves in higherdimensional spaces7.1 Basic definitions and properties; 7.2 Rational transformations; 7.3 Hurwitz's Theorem; 7.4 Linear series composed of an involution; 7.5 The canonical curve; 7.6 Osculating hyperplanes and ramification divisors; 7.7 Nonclassical curves and linear systems of lines; 7.8 Nonclassical curves and linear systems of conics; 7.9 Dual curves of space curves; 7.10 Complete linear series of small order; 7.11 Examples of curves; 7.12 The Linear General Position Principle; 7.13 Castelnuovo's Bound; 7.14 A generalisation of Clifford's Theorem
 7.15 The Uniform Position Principle7.16 Valuation rings; 7.17 Curves as algebraic varieties of dimension one; 7.18 Exercises; 7.19 Notes; PART 2. CURVES OVER A FINITE FIELD; Chapter 8. Rational points and places over a finite field; 8.1 Plane curves defined over a finite field; 8.2 Fqrational branches of a curve; 8.3 Fqrational places, divisors and linear series; 8.4 Space curves over Fq; 8.5 The StöhrVoloch Theorem; 8.6 Frobenius classicality with respect to lines; 8.7 Frobenius classicality with respect to conics; 8.8 The dual of a Frobenius nonclassical curve; 8.9 Exercises; 8.10 Notes
 Isbn
 9781400847419
 Label
 Algebraic curves over a finite field
 Title
 Algebraic curves over a finite field
 Statement of responsibility
 J.W.P. Hirschfeld, G. Korchmaros, F. Torres
 Language
 eng
 Summary
 This book provides an accessible and selfcontained introduction to the theory of algebraic curves over a finite field, a subject that has been of fundamental importance to mathematics for many years and that has essential applications in areas such as finite geometry, number theory, errorcorrecting codes, and cryptology. Unlike other books, this one emphasizes the algebraic geometry rather than the function field approach to algebraic curves. The authors begin by developing the general theory of curves over any field, highlighting peculiarities occurring for positive characteristi
 Cataloging source
 E7B
 Index
 index present
 Language note
 In English
 Literary form
 non fiction
 Nature of contents

 dictionaries
 bibliography
 Series statement
 Princeton Series in Applied Mathematics
 Label
 Algebraic curves over a finite field, J.W.P. Hirschfeld, G. Korchmaros, F. Torres
 Bibliography note
 Includes bibliographical references and index
 Carrier category
 online resource
 Carrier category code
 cr
 Carrier MARC source
 rdacarrier
 Color
 multicolored
 Content category
 text
 Content type code
 txt
 Content type MARC source
 rdacontent
 Contents

 Cover; Title; Copyright; Dedication; Contents; Preface; PART 1. GENERAL THEORY OF CURVES; Chapter 1. Fundamental ideas; 1.1 Basic definitions; 1.2 Polynomials; 1.3 Affine plane curves; 1.4 Projective plane curves; 1.5 The Hessian curve; 1.6 Projective varieties in higherdimensional spaces; 1.7 Exercises; 1.8 Notes; Chapter 2. Elimination theory; 2.1 Elimination of one unknown; 2.2 The discriminant; 2.3 Elimination in a system in two unknowns; 2.4 Exercises; 2.5 Notes; Chapter 3. Singular points and intersections; 3.1 The intersection number of two curves; 3.2 Bézout's Theorem
 3.3 Rational and birational transformations3.4 Quadratic transformations; 3.5 Resolution of singularities; 3.6 Exercises; 3.7 Notes; Chapter 4. Branches and parametrisation; 4.1 Formal power series; 4.2 Branch representations; 4.3 Branches of plane algebraic curves; 4.4 Local quadratic transformations; 4.5 Noether's Theorem; 4.6 Analytic branches; 4.7 Exercises; 4.8 Notes; Chapter 5. The function field of a curve; 5.1 Generic points; 5.2 Rational transformations; 5.3 Places; 5.4 Zeros and poles; 5.5 Separability and inseparability; 5.6 Frobenius rational transformations
 5.7 Derivations and differentials5.8 The genus of a curve; 5.9 Residues of differential forms; 5.10 Higher derivatives in positive characteristic; 5.11 The dual and bidual of a curve; 5.12 Exercises; 5.13 Notes; Chapter 6. Linear series and the RiemannRoch Theorem; 6.1 Divisors and linear series; 6.2 Linear systems of curves; 6.3 Special and nonspecial linear series; 6.4 Reformulation of the RiemannRoch Theorem; 6.5 Some consequences of the RiemannRoch Theorem; 6.6 The Weierstrass Gap Theorem; 6.7 The structure of the divisor class group; 6.8 Exercises; 6.9 Notes
 Chapter 7. Algebraic curves in higherdimensional spaces7.1 Basic definitions and properties; 7.2 Rational transformations; 7.3 Hurwitz's Theorem; 7.4 Linear series composed of an involution; 7.5 The canonical curve; 7.6 Osculating hyperplanes and ramification divisors; 7.7 Nonclassical curves and linear systems of lines; 7.8 Nonclassical curves and linear systems of conics; 7.9 Dual curves of space curves; 7.10 Complete linear series of small order; 7.11 Examples of curves; 7.12 The Linear General Position Principle; 7.13 Castelnuovo's Bound; 7.14 A generalisation of Clifford's Theorem
 7.15 The Uniform Position Principle7.16 Valuation rings; 7.17 Curves as algebraic varieties of dimension one; 7.18 Exercises; 7.19 Notes; PART 2. CURVES OVER A FINITE FIELD; Chapter 8. Rational points and places over a finite field; 8.1 Plane curves defined over a finite field; 8.2 Fqrational branches of a curve; 8.3 Fqrational places, divisors and linear series; 8.4 Space curves over Fq; 8.5 The StöhrVoloch Theorem; 8.6 Frobenius classicality with respect to lines; 8.7 Frobenius classicality with respect to conics; 8.8 The dual of a Frobenius nonclassical curve; 8.9 Exercises; 8.10 Notes
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 {'f': 'http://opac.lib.rpi.edu/record=b4332083'}
 Extent
 1 online resource (717 pages).
 Form of item
 online
 Isbn
 9781400847419
 Media category
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 Media MARC source
 rdamedia
 Media type code
 c
 Specific material designation
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<div class="citation" vocab="http://schema.org/"><i class="fa faexternallinksquare fafw"></i> Data from <span resource="http://link.lib.rpi.edu/portal/AlgebraiccurvesoverafinitefieldJ.W.P./sXPNStKXQbA/" typeof="WorkExample http://bibfra.me/vocab/lite/Item"><span property="name http://bibfra.me/vocab/lite/label"><a href="http://link.lib.rpi.edu/portal/AlgebraiccurvesoverafinitefieldJ.W.P./sXPNStKXQbA/">Algebraic curves over a finite field, J.W.P. Hirschfeld, G. Korchmaros, F. Torres</a></span>  <span property="offers" typeOf="Offer"><span property="offeredBy" typeof="Library ll:Library" resource="http://link.lib.rpi.edu/"><span property="name http://bibfra.me/vocab/lite/label"><a property="url" href="http://link.lib.rpi.edu/">Rensselaer Libraries</a></span></span></span></span></div>