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

• Autore: Hirschfeld J. W. P (James William Peter), <1940->
• 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
• 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
• Serie: Princeton series in applied mathematics.