RATIONAL CURVES AND CONICS

DETAILS

 
Introduction

 
Rational Curves

      Examples of Curves
            Example CrvCon_rational_curve (H84E1)
            Example CrvCon_conic_curve (H84E2)

      Parametrisation of Rational Curves
            RationalCurve(X,f) : Sch,RngMPolElt -> CrvRat
            IsRationalCurve(X) : Sch -> BoolElt,CrvRat
            Parametrisation(X,C,p) : Sch,Crv,Pt -> MapSch
            Example CrvCon_rat_crv_parametrisation (H84E3)
            Example CrvCon_aff-param (H84E4)

 
Conic Curves

      Construction of a Conic
            Conic(X,f:parameters) : Sch, RngMPolElt -> CrvCon
            IsConic(X) : Sch -> BoolElt, CrvCon
            Example CrvCon_Creation (H84E5)

      Access Functions
            Polynomial(C) : CrvCon -> RngMPolElt
            LegendreEquation(C) : CrvCon -> RngMPolElt, ModMatRngElt
            ReducedForm(C) : CrvCon -> RngMPolElt, ModMatRngElt
            HasSolubilityCertificate(C) : CrvCon -> BoolElt, SeqEnum
            HasSolubilityCertificate(S) : SeqEnum[RngIntElt] -> BoolElt, SeqEnum
            SolubilityCertificate(C) : CrvCon -> SeqEnum
            Example CrvCon_Access (H84E6)

 
Finding Points on Conics

      Finding a Reduced Solution
            HasPoint(C) : CrvCon -> BoolElt, Pt
            ReducedProjectiveSolution(C) : CrvCon -> Pt
            HasReducedAffinePoint(C) : CrvCon -> BoolElt, Pt
            ReducedAffineSolution(C) : CrvCon -> BoolElt, Pt

      All Solutions via Parametrization
            ParametrizationMatrix(C) : CrvCon -> ModMatRngElt
            Parametrization(X,C) : Sch,CrvCon -> MapSch
            Solution(C) : CrvCon -> Pt

      Reducing a Point
            Reduction(p: parameters) : Pt -> Pt
            IsReduced(p) : Pt -> BoolElt
            Example CrvCon_Solution (H84E7)

 
Bibliography