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, RngUPolElt, RngUPolElt -> FldFunRatElt
            F ! a : FldFunRat, FldElt -> FldFunRatElt
            K . i : FldFunRat, RngIntElt -> FldFunRatElt
            Example FldFunRat_FunctionField (H45E1)

 
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, y₁, ..., y_n > : FldFunRat, Rng -> Map
            Example FldFunRat_Homomorphism (H45E2)

 
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
            Numerator(f, R) : FldFunRatElt -> RngElt

      Predicates on Ring Elements

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

      Decomposition
            Decomposition(f) : FldFunRatUElt -> [[FldFunRatUElt]]
            Example FldFunRat_decomp-ex (H45E3)

      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 (H45E4)

 
Padé-Hermite Approximants

      Introduction

      Ordering of Sequences
            MaximumDegree(f) : SeqEnum -> RngIntElt
            Example FldFunRat_degree-of-sequence (H45E5)
            TypeOfSequence(f) : SeqEnum -> RngIntElt, RngIntElt
            Example FldFunRat_type-of-sequence (H45E6)
            MinimalVectorSequence(f,n) : SeqEnum, RngIntElt -> SeqEnum
            Example FldFunRat_minimal-vector-sequence (H45E7)
            Example FldFunRat_the-next_example (H45E8)
            Example FldFunRat_another-example (H45E9)
            Example FldFunRat_one-more (H45E10)

      Approximants
            PadeHermiteApproximant(f,d) : SeqEnum, SeqEnum -> ModTupRngElt, SeqEnum, RngIntElt
            Example FldFunRat_pade-hermite-approximants (H45E11)
            Example FldFunRat_last-example (H45E12)
            Example FldFunRat_ (H45E13)
            PadeHermiteApproximant(f,m) : SeqEnum, RngIntElt -> ModTupRngElt, SeqEnum
            Example FldFunRat_pade-hermite-approximants-vectors (H45E14)
            Example FldFunRat_ (H45E15)

 
Bibliography

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