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Algebraic curves over a finite field / / J. W. P. Hirschfeld, G. Korchmaros, F. Torres



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Autore: Hirschfeld J. W. P (James William Peter), <1940-> Visualizza persona
Titolo: Algebraic curves over a finite field / / J. W. P. Hirschfeld, G. Korchmaros, F. Torres Visualizza cluster
Pubblicazione: Princeton, New Jersey : , : Princeton University Press, , 2008
©2008
Edizione: Course Book
Descrizione fisica: 1 online resource (717 p.)
Disciplina: 516.352
Soggetto topico: Curves, Algebraic
Finite fields (Algebra)
Soggetto non controllato: Abelian group
Abelian variety
Affine plane
Affine space
Affine variety
Algebraic closure
Algebraic curve
Algebraic equation
Algebraic extension
Algebraic function
Algebraic geometry
Algebraic integer
Algebraic number field
Algebraic number theory
Algebraic number
Algebraic variety
Algebraically closed field
Applied mathematics
Automorphism
Birational invariant
Characteristic exponent
Classification theorem
Clifford's theorem
Combinatorics
Complex number
Computation
Cyclic group
Cyclotomic polynomial
Degeneracy (mathematics)
Degenerate conic
Divisor (algebraic geometry)
Divisor
Dual curve
Dual space
Elliptic curve
Equation
Fermat curve
Finite field
Finite geometry
Finite group
Formal power series
Function (mathematics)
Function field
Fundamental theorem
Galois extension
Galois theory
Gauss map
General position
Generic point
Geometry
Homogeneous polynomial
Hurwitz's theorem
Hyperelliptic curve
Hyperplane
Identity matrix
Inequality (mathematics)
Intersection number (graph theory)
Intersection number
J-invariant
Line at infinity
Linear algebra
Linear map
Mathematical induction
Mathematics
Menelaus' theorem
Modular curve
Natural number
Number theory
Parity (mathematics)
Permutation group
Plane curve
Point at infinity
Polar curve
Polygon
Polynomial
Power series
Prime number
Projective plane
Projective space
Quadratic transformation
Quadric
Resolution of singularities
Riemann hypothesis
Scalar multiplication
Scientific notation
Separable extension
Separable polynomial
Sign (mathematics)
Singular point of a curve
Special case
Subgroup
Sylow theorems
System of linear equations
Tangent
Theorem
Transcendence degree
Upper and lower bounds
Valuation ring
Variable (mathematics)
Vector space
Classificazione: SK 240
Persona (resp. second.): KorchmárosG
TorresF (Fernando)
Note generali: Description based upon print version of record.
Nota di bibliografia: Includes bibliographical references and index.
Nota di contenuto: Front matter -- Contents -- Preface -- PART 1. General theory of curves -- Chapter One. Fundamental ideas -- Chapter Two. Elimination theory -- Chapter Three. Singular points and intersections -- Chapter Four. Branches and parametrisation -- Chapter Five. The function field of a curve -- Chapter Six. Linear series and the Riemann-Roch Theorem -- Chapter Seven. Algebraic curves in higher-dimensional spaces -- PART 2. Curves over a finite field -- Chapter Eight. Rational points and places over a finite field -- Chapter Nine. Zeta functions and curves with many rational points -- PART 3. Further developments -- Chapter Ten. Maximal and optimal curves -- Chapter Eleven. Automorphisms of an algebraic curve -- Chapter Twelve. Some families of algebraic curves -- Chapter Thirteen. Applications: codes and arcs -- Appendix A. Background on field theory and group theory -- Appendix B. Notation -- Bibliography -- Index
Sommario/riassunto: This book provides an accessible and self-contained 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, error-correcting 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 characteristic and requiring of the reader only basic knowledge of algebra and geometry. The special properties that a curve over a finite field can have are then discussed. The geometrical theory of linear series is used to find estimates for the number of rational points on a curve, following the theory of Stöhr and Voloch. The approach of Hasse and Weil via zeta functions is explained, and then attention turns to more advanced results: a state-of-the-art introduction to maximal curves over finite fields is provided; a comprehensive account is given of the automorphism group of a curve; and some applications to coding theory and finite geometry are described. The book includes many examples and exercises. It is an indispensable resource for researchers and the ideal textbook for graduate students.
Titolo autorizzato: Algebraic curves over a finite field  Visualizza cluster
ISBN: 1-4008-4741-9
Formato: Materiale a stampa
Livello bibliografico Monografia
Lingua di pubblicazione: Inglese
Record Nr.: 9910812510203321
Lo trovi qui: Univ. Federico II
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Serie: Princeton series in applied mathematics.