RATIONAL FUNCTION FIELDS

DETAILS

 
Introduction

 
Creation Functions

      Creation of Structures
            FunctionField(R) : Rng -> FldFunRat
            FunctionField(R, r) : Rng, RngIntElt -> FldFunRat
            FieldOfFractions(P) : RngUPol -> FldFunRat

      Names
            AssignNames(~F, s) : FldFunRat, [ MonStgElt ]) ->
            Name(F, i) : FldFunRat, RngIntElt -> FldFunRatElt

      Creation of Elements
            F ! [a, b] : FldFunRat, RngPolElt, RngPolElt -> FldFunRatElt
            F ! a : FldFunRat, FldElt -> FldFunRatElt
            K . i : FldFunRat, RngIntElt -> FldFunRatElt
            Example FldFunRat_FunctionField (H54E1)

 
Structure Operations

      Related Structures
            IntegerRing(F) : FldFunRat -> RngPol
            BaseRing(F) : FldFunRat -> Rng
            Rank(F) : FldFunRat -> RngIntElt
            ValuationRing(F) : FldFunRat -> RngVal
            ValuationRing(F, f) : FldFunRat, RngUPolElt -> RngVal

      Invariants

      Ring Predicates and Booleans

      Homomorphisms
            hom< P -> S | f, y_1, ..., y_n > : FldFunRat, Rng -> Map
            Example FldFunRat_Homomorphism (H54E2)

 
Element Operations

      Arithmetic

      Equality and Membership

      Numerator, Denominator and Degree
            Numerator(f) : FldFunRatElt -> RngElt
            Denominator(f) : FldFunRatElt -> RngElt
            Degree(f) : FldFunRatElt -> RngIntElt
            TotalDegree(f) : FldFunRatElt -> RngIntElt
            WeightedDegree(f) : FldFunRatElt -> RngIntElt

      Predicates on Ring Elements

      Evaluation
            Evaluate(f, r) : FldFunRatUElt, RngElt -> FldFunRatUElt
            Evaluate(f, v, r) : FldFunRatMElt, RngIntElt, RngElt -> FldFunRatMElt

      Derivative
            Derivative(f) : FldFunRatUElt -> FldFunRatUElt
            Derivative(f, k) : FldFunRatUElt, RngIntElt -> FldFunRatUElt
            Derivative(f, v) : FldFunRatMElt, RngIntElt -> FldFunRatMElt
            Derivative(f, v, k) : FldFunRatMElt, RngIntElt, RngIntElt -> FldFunRatMElt

      Partial Fraction Decomposition
            PartialFractionDecomposition(f) : FldFunRatUElt -> [ <RngUPolElt, RngIntElt, RngUPolElt> ]
            SquarefreePartialFractionDecomposition(f) : FldFunRatUElt -> [ <RngUPolElt, RngIntElt, RngUPolElt> ]
            Example FldFunRat_PartialFractionDecomposition (H54E3)