LEADER 07832nam 2201897 450 001 9910786795203321 005 20210701014950.0 010 $a1-4008-4741-9 024 7 $a10.1515/9781400847419 035 $a(CKB)3710000000202214 035 $a(EBL)1138042 035 $a(OCoLC)884645547 035 $a(SSID)ssj0001261434 035 $a(PQKBManifestationID)11838488 035 $a(PQKBTitleCode)TC0001261434 035 $a(PQKBWorkID)11320440 035 $a(PQKB)10917622 035 $a(DE-B1597)453495 035 $a(OCoLC)889240929 035 $a(OCoLC)979727067 035 $a(DE-B1597)9781400847419 035 $a(Au-PeEL)EBL1138042 035 $a(CaPaEBR)ebr10901634 035 $a(CaONFJC)MIL630111 035 $a(MiAaPQ)EBC1138042 035 $a(PPN)201964074 035 $a(EXLCZ)993710000000202214 100 $a20140813h20082008 uy 0 101 0 $aeng 135 $aurnn#---||||| 181 $ctxt 182 $cc 183 $acr 200 10$aAlgebraic curves over a finite field /$fJ. W. P. Hirschfeld, G. Korchmaros, F. Torres 205 $aCourse Book 210 1$aPrinceton, New Jersey :$cPrinceton University Press,$d2008. 210 4$d©2008 215 $a1 online resource (717 p.) 225 1 $aPrinceton Series in Applied Mathematics 300 $aDescription based upon print version of record. 311 0 $a0-691-09679-1 320 $aIncludes bibliographical references and index. 327 $tFront matter --$tContents --$tPreface --$tPART 1. General theory of curves --$tChapter One. Fundamental ideas --$tChapter Two. Elimination theory --$tChapter Three. Singular points and intersections --$tChapter Four. Branches and parametrisation --$tChapter Five. The function field of a curve --$tChapter Six. Linear series and the Riemann-Roch Theorem --$tChapter Seven. Algebraic curves in higher-dimensional spaces --$tPART 2. Curves over a finite field --$tChapter Eight. Rational points and places over a finite field --$tChapter Nine. Zeta functions and curves with many rational points --$tPART 3. Further developments --$tChapter Ten. Maximal and optimal curves --$tChapter Eleven. Automorphisms of an algebraic curve --$tChapter Twelve. Some families of algebraic curves --$tChapter Thirteen. Applications: codes and arcs --$tAppendix A. Background on field theory and group theory --$tAppendix B. Notation --$tBibliography --$tIndex 330 $aThis book provides an accessible and self-contained introduction to the theory of algebraic curves over a finite field, a subject that has been of fundamental importance to mathematics for many years and that has essential applications in areas such as finite geometry, number theory, error-correcting codes, and cryptology. Unlike other books, this one emphasizes the algebraic geometry rather than the function field approach to algebraic curves. The authors begin by developing the general theory of curves over any field, highlighting peculiarities occurring for positive characteristic and requiring of the reader only basic knowledge of algebra and geometry. The special properties that a curve over a finite field can have are then discussed. The geometrical theory of linear series is used to find estimates for the number of rational points on a curve, following the theory of Stöhr and Voloch. The approach of Hasse and Weil via zeta functions is explained, and then attention turns to more advanced results: a state-of-the-art introduction to maximal curves over finite fields is provided; a comprehensive account is given of the automorphism group of a curve; and some applications to coding theory and finite geometry are described. The book includes many examples and exercises. It is an indispensable resource for researchers and the ideal textbook for graduate students. 410 0$aPrinceton series in applied mathematics. 606 $aCurves, Algebraic 606 $aFinite fields (Algebra) 610 $aAbelian group. 610 $aAbelian variety. 610 $aAffine plane. 610 $aAffine space. 610 $aAffine variety. 610 $aAlgebraic closure. 610 $aAlgebraic curve. 610 $aAlgebraic equation. 610 $aAlgebraic extension. 610 $aAlgebraic function. 610 $aAlgebraic geometry. 610 $aAlgebraic integer. 610 $aAlgebraic number field. 610 $aAlgebraic number theory. 610 $aAlgebraic number. 610 $aAlgebraic variety. 610 $aAlgebraically closed field. 610 $aApplied mathematics. 610 $aAutomorphism. 610 $aBirational invariant. 610 $aCharacteristic exponent. 610 $aClassification theorem. 610 $aClifford's theorem. 610 $aCombinatorics. 610 $aComplex number. 610 $aComputation. 610 $aCyclic group. 610 $aCyclotomic polynomial. 610 $aDegeneracy (mathematics). 610 $aDegenerate conic. 610 $aDivisor (algebraic geometry). 610 $aDivisor. 610 $aDual curve. 610 $aDual space. 610 $aElliptic curve. 610 $aEquation. 610 $aFermat curve. 610 $aFinite field. 610 $aFinite geometry. 610 $aFinite group. 610 $aFormal power series. 610 $aFunction (mathematics). 610 $aFunction field. 610 $aFundamental theorem. 610 $aGalois extension. 610 $aGalois theory. 610 $aGauss map. 610 $aGeneral position. 610 $aGeneric point. 610 $aGeometry. 610 $aHomogeneous polynomial. 610 $aHurwitz's theorem. 610 $aHyperelliptic curve. 610 $aHyperplane. 610 $aIdentity matrix. 610 $aInequality (mathematics). 610 $aIntersection number (graph theory). 610 $aIntersection number. 610 $aJ-invariant. 610 $aLine at infinity. 610 $aLinear algebra. 610 $aLinear map. 610 $aMathematical induction. 610 $aMathematics. 610 $aMenelaus' theorem. 610 $aModular curve. 610 $aNatural number. 610 $aNumber theory. 610 $aParity (mathematics). 610 $aPermutation group. 610 $aPlane curve. 610 $aPoint at infinity. 610 $aPolar curve. 610 $aPolygon. 610 $aPolynomial. 610 $aPower series. 610 $aPrime number. 610 $aProjective plane. 610 $aProjective space. 610 $aQuadratic transformation. 610 $aQuadric. 610 $aResolution of singularities. 610 $aRiemann hypothesis. 610 $aScalar multiplication. 610 $aScientific notation. 610 $aSeparable extension. 610 $aSeparable polynomial. 610 $aSign (mathematics). 610 $aSingular point of a curve. 610 $aSpecial case. 610 $aSubgroup. 610 $aSylow theorems. 610 $aSystem of linear equations. 610 $aTangent. 610 $aTheorem. 610 $aTranscendence degree. 610 $aUpper and lower bounds. 610 $aValuation ring. 610 $aVariable (mathematics). 610 $aVector space. 615 0$aCurves, Algebraic. 615 0$aFinite fields (Algebra) 676 $a516.352 686 $aSK 240$2rvk 700 $aHirschfeld$b J. W. P$g(James William Peter),$f1940-$01522609 702 $aKorchma?ros$b G. 702 $aTorres$b F$g(Fernando), 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910786795203321 996 $aAlgebraic curves over a finite field$93762386 997 $aUNINA