Rational Function Fields

• Introduction
• Creation Functions
  • Creation of Structures
    • FunctionField(R): Rng → FldFunRat
    • RationalFunctionField(R): Rng → FldFunRat
    • FunctionField(R, r): Rng, RngIntElt → FldFunRat
    • RationalFunctionField(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
    • elt< F | a, b >: FldFunRat, RngUPolElt, RngUPolElt → FldFunRatElt
    • F ! a: FldFunRat, FldElt → FldFunRatElt
    • K . i: FldFunRat, RngIntElt → FldFunRatElt
    • One(F): FldFunRat → FldFunRatElt
    • Identity(F): FldFunRat → FldFunRatElt
    • Zero(F): FldFunRat → FldFunRatElt
    • Representative(F): FldFunRat → FldFunRatElt
    • Example: Function Field
• Structure Operations
  • Related Structures
    • IntegerRing(F): FldFunRat → RngPol
    • RingOfIntegers(F): FldFunRat → RngPol
    • BaseRing(F): FldFunRat → Rng
    • CoefficientRing(F): FldFunRat → Rng
    • Rank(F): FldFunRat → RngIntElt
    • ValuationRing(F): FldFunRat → RngVal
    • ValuationRing(F, f): FldFunRat, RngUPolElt → RngVal
    • Category(R): FldFunRat → Cat
    • Parent(R): FldFunRat → Pow
    • PrimeRing(R): FldFunRat → Rng
  • Invariants
    • Characteristic(F): FldFunRat → FldFunRatElt
  • Ring Predicates and Booleans
    • IsCommutative(F): FldFunRat → BoolElt
    • IsUnitary(F): FldFunRat → BoolElt
    • IsFinite(F): FldFunRat → BoolElt
    • IsOrdered(F): FldFunRat → BoolElt
    • IsField(F): FldFunRat → BoolElt
    • IsEuclideanDomain(F): FldFunRat → BoolElt
    • IsPID(F): FldFunRat → BoolElt
    • IsUFD(F): FldFunRat → BoolElt
    • IsDivisionRing(F): FldFunRat → BoolElt
    • IsEuclideanRing(F): FldFunRat → BoolElt
    • IsPrincipalIdealRing(F): FldFunRat → BoolElt
    • IsDomain(F): FldFunRat → BoolElt
    • F eq G: FldFunRat, Rng → BoolElt
    • F ne G: FldFunRat, Rng → BoolElt
  • Homomorphisms
    • hom< P -> S | f, y₁, ..., yₙ >: FldFunRat, Rng → Map
    • hom< P -> S | y₁, ..., yₙ >: FldFunRat, Rng → Map
    • Example: Homomorphism
• Element Operations
  • Arithmetic
    • + a: FldFunRatElt → FldFunRatElt
    • - a: FldFunRatElt → FldFunRatElt
    • a + b: FldFunRatElt, FldFunRatElt → FldFunRatElt
    • a - b: FldFunRatElt, FldFunRatElt → FldFunRatElt
    • a * b: FldFunRatElt, FldFunRatElt → FldFunRatElt
    • a / b: FldFunRatElt, FldFunRatElt → FldFunRatElt
    • a ^ k: FldFunRatElt, RngIntElt → FldFunRatElt
  • Equality and Membership
    • a eq b: FldFunRatElt, FldFunRatElt → BoolElt
    • a ne b: FldFunRatElt, FldFunRatElt → BoolElt
    • a in F: FldFunRatElt, FldFunRat → BoolElt
    • a notin F: FldFunRatElt, FldFunRat → BoolElt
  • Numerator, Denominator and Degree
    • Numerator(f): FldFunRatElt → RngElt
    • Numerator(f): RngUPolElt → RngUPolElt
    • Numerator(f): RngMPolElt → RngMPolElt
    • Denominator(f): FldFunRatElt → RngElt
    • Denominator(f): RngUPolElt → RngUPolElt
    • Denominator(f): RngMPolElt → RngMPolElt
    • Degree(f): FldFunRatElt → RngIntElt
    • TotalDegree(f): FldFunRatElt → RngIntElt
    • WeightedDegree(f): FldFunRatElt → RngIntElt
    • Numerator(f, R): FldFunRatElt → RngElt
    • Denominator(f, R): FldFunRatElt → RngElt
  • Predicates on Ring Elements
    • IsZero(a): FldFunRatElt → BoolElt
    • IsOne(a): FldFunRatElt → BoolElt
    • IsMinusOne(a): FldFunRatElt → BoolElt
    • IsNilpotent(a): FldFunRatElt → BoolElt
    • IsIdempotent(a): FldFunRatElt → BoolElt
    • IsUnit(a): FldFunRatElt → BoolElt
    • IsZeroDivisor(a): FldFunRatElt → BoolElt
    • IsRegular(a): FldFunRatElt → BoolElt
  • 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: Decomp Ex
  • 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: Partial Fraction Decomposition
• Padé-Hermite Approximants
  • Introduction
  • Ordering of Sequences
    • MaximumDegree(f): SeqEnum → RngIntElt
    • Example: Degree Of Sequence
    • TypeOfSequence(f): SeqEnum → RngIntElt, RngIntElt
    • Example: Type Of Sequence
    • MinimalVectorSequence(f,n): SeqEnum, RngIntElt → SeqEnum
    • Example: Minimal Vector Sequence
    • Example: The Next Example
    • Example: Another Example
    • Example: One More
  • Approximants
    • PadeHermiteApproximant(f,d): SeqEnum, SeqEnum → ModTupRngElt, SeqEnum, RngIntElt
    • Example: Pade Hermite Approximants
    • Example: Last Example
    • Example
    • PadeHermiteApproximant(f,m): SeqEnum, RngIntElt → ModTupRngElt, SeqEnum
    • Example: Pade Hermite Approximants Vectors
    • Example