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RATIONAL FUNCTION FIELDS

 
Acknowledgements
 
Introduction
 
Creation Functions
      Creation of Structures
      Names
      Creation of Elements
 
Structure Operations
      Related Structures
      Invariants
      Ring Predicates and Booleans
      Homomorphisms
 
Element Operations
      Arithmetic
      Equality and Membership
      Numerator, Denominator and Degree
      Predicates on Ring Elements
      Evaluation
      Derivative
      Partial Fraction Decomposition
 
Padé-Hermite Approximants
      Introduction
      Ordering of Sequences
      Approximants
 
Bibliography







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, RngUPolElt, RngUPolElt -> FldFunRatElt
            F ! a : FldFunRat, FldElt -> FldFunRatElt
            K . i : FldFunRat, RngIntElt -> FldFunRatElt
            Example FldFunRat_FunctionField (H41E1)

 
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, ..., yn > : FldFunRat, Rng -> Map
            Example FldFunRat_Homomorphism (H41E2)

 
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 (H41E3)

 
Padé-Hermite Approximants

      Introduction

      Ordering of Sequences
            MaximumDegree(f) : SeqEnum -> RngIntElt
            Example FldFunRat_degree-of-sequence (H41E4)
            TypeOfSequence(f) : SeqEnum -> RngIntElt, RngIntElt
            Example FldFunRat_type-of-sequence (H41E5)
            MinimalVectorSequence(f,n) : SeqEnum, RngIntElt -> SeqEnum
            Example FldFunRat_minimal-vector-sequence (H41E6)
            Example FldFunRat_the-next_example (H41E7)
            Example FldFunRat_another-example (H41E8)
            Example FldFunRat_one-more (H41E9)

      Approximants
            PadeHermiteApproximant(f,d) : SeqEnum, SeqEnum -> ModTupRngElt, SeqEnum, RngIntElt
            Example FldFunRat_pade-hermite-approximants (H41E10)
            Example FldFunRat_last-example (H41E11)
            Example FldFunRat_ (H41E12)
            PadeHermiteApproximant(f,m) : SeqEnum, RngIntElt -> ModTupRngElt, SeqEnum
            Example FldFunRat_pade-hermite-approximants-vectors (H41E13)
            Example FldFunRat_ (H41E14)

 
Bibliography

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