Rational Function Fields
- Introduction
- Creation Functions
- Creation of Structures
- Names
- Creation of Elements
F ! [a, b]: FldFunRat, RngUPolElt, RngUPolElt → FldFunRatEltelt< F | a, b >: FldFunRat, RngUPolElt, RngUPolElt → FldFunRatEltF ! a: FldFunRat, FldElt → FldFunRatEltK . i: FldFunRat, RngIntElt → FldFunRatEltOne(F): FldFunRat → FldFunRatEltIdentity(F): FldFunRat → FldFunRatEltZero(F): FldFunRat → FldFunRatEltRepresentative(F): FldFunRat → FldFunRatElt- Example: Creating Rational Function Field Elements
- Structure Operations
- Related Structures
IntegerRing(F): FldFunRat → RngPolRingOfIntegers(F): FldFunRat → RngPolBaseRing(F): FldFunRat → RngCoefficientRing(F): FldFunRat → RngRank(F): FldFunRat → RngIntEltValuationRing(F): FldFunRat → RngValValuationRing(F, f): FldFunRat, RngUPolElt → RngValCategory(R): FldFunRat → CatParent(R): FldFunRat → PowPrimeRing(R): FldFunRat → Rng
- Invariants
- Ring Predicates and Booleans
IsCommutative(F): FldFunRat → BoolEltIsUnitary(F): FldFunRat → BoolEltIsFinite(F): FldFunRat → BoolEltIsOrdered(F): FldFunRat → BoolEltIsField(F): FldFunRat → BoolEltIsEuclideanDomain(F): FldFunRat → BoolEltIsPID(F): FldFunRat → BoolEltIsUFD(F): FldFunRat → BoolEltIsDivisionRing(F): FldFunRat → BoolEltIsEuclideanRing(F): FldFunRat → BoolEltIsPrincipalIdealRing(F): FldFunRat → BoolEltIsDomain(F): FldFunRat → BoolEltF eq G: FldFunRat, Rng → BoolEltF ne G: FldFunRat, Rng → BoolElt
- Homomorphisms
- Related Structures
- Element Operations
- Arithmetic
+ a: FldFunRatElt → FldFunRatElt- a: FldFunRatElt → FldFunRatElta + b: FldFunRatElt, FldFunRatElt → FldFunRatElta - b: FldFunRatElt, FldFunRatElt → FldFunRatElta * b: FldFunRatElt, FldFunRatElt → FldFunRatElta / b: FldFunRatElt, FldFunRatElt → FldFunRatElta ^ k: FldFunRatElt, RngIntElt → FldFunRatElt
- Equality and Membership
- Numerator, Denominator, and Degree
Numerator(f): FldFunRatElt → RngEltNumerator(f): RngUPolElt → RngUPolEltNumerator(f): RngMPolElt → RngMPolEltDenominator(f): FldFunRatElt → RngEltDenominator(f): RngUPolElt → RngUPolEltDenominator(f): RngMPolElt → RngMPolEltDegree(f): FldFunRatElt → RngIntEltTotalDegree(f): FldFunRatElt → RngIntEltWeightedDegree(f): FldFunRatElt → RngIntEltNumerator(f, R): FldFunRatElt → RngEltDenominator(f, R): FldFunRatElt → RngElt
- Predicates on Ring Elements
- Evaluation
- Decomposition
- Derivative
- Partial Fraction Decomposition
- Arithmetic
- Padé-Hermite Approximants
- Introduction
- Ordering of Sequences
MaximumDegree(f): SeqEnum → RngIntElt- Example: Maximum Degree of a Sequence
TypeOfSequence(f): SeqEnum → RngIntElt, RngIntElt- Example: Type of a Sequence
MinimalVectorSequence(f,n): SeqEnum, RngIntElt → SeqEnum- Example: Minimal Vector Sequence of Power Series
- Example: Minimal Vector Sequence of Polynomials
- Example: Minimal Vector Sequence of Higher Order
- Example: Minimal Vector Sequence with Distortion
- Approximants
PadeHermiteApproximant(f,d): SeqEnum, SeqEnum → ModTupRngElt, SeqEnum, RngIntElt- Example: Pade-Hermite Approximants
- Example: Pade-Hermite Approximant with Distortion
- Example: Degrees and Valuations of Approximants
PadeHermiteApproximant(f,m): SeqEnum, RngIntElt → ModTupRngElt, SeqEnum- Example: Pade-Hermite Approximants of Vectors
- Example: Approximants of Sine, Cosine, and Exponential
- Bibliography