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AFFINE ALGEBRAS

 
Acknowledgements
 
Introduction
 
Creation of Affine Algebras
 
Operations on Affine Algebras
 
Maps between Affine Algebras
 
Finite Dimensional Affine Algebras
 
Affine Algebras which are Fields
 
Rings and Fields of Fractions of Affine Algebras







DETAILS

 
Introduction

 
Creation of Affine Algebras
      quo< P | J > : RngMPol, RngMPol -> RngMPolRes
      P / J : RngMPol, RngMPol -> RngMPolRes
      AffineAlgebra< R, X | L > : Fld, List, List -> RngMPolRes
      Example AlgAff_Creation (H108E1)

 
Operations on Affine Algebras
      Q . i : RngMPolRes, RngIntElt -> RngMPolResElt
      CoefficientRing(Q) : RngMPolRes -> Rng
      Rank(Q) : RngMPolRes -> RngIntElt
      DivisorIdeal(I) : RngMPolRes -> RngMPol
      PreimageIdeal(I) : RngMPolRes -> RngMPol
      PreimageRing(Q) : RngMPolRes -> RngMPol
      OriginalRing(Q) : RngMPolRes -> Rng
      I eq J : RngMPolRes, RngMPolRes -> BoolElt
      I subset J : RngMPolRes, RngMPolRes -> BoolElt
      I + J : RngMPolRes, RngMPolRes -> RngMPolRes
      I * J : RngMPolRes, RngMPolRes -> RngMPolRes
      I ^ n : RngMPolRes, RngIntElt -> BoolElt
      I meet J : RngMPolRes, RngMPolRes -> RngMPolRes
      IsProper(I) : RngMPolRes -> BoolElt
      IsZero(I) : RngMPolRes -> BoolElt
      IsPrime(I) : RngMPolRes -> BoolElt
      IsPrimary(I) : RngMPolRes -> BoolElt
      IsRadical(I) : RngMPolRes -> BoolElt
      PrimaryDecomposition(I) : RngMPolRes -> [ RngMPolRes ], [ RngMPolRes ]
      RadicalDecomposition(I) : RngMPolRes -> [ RngMPolRes ]
      Example AlgAff_Operations (H108E2)

 
Maps between Affine Algebras
      AffineAlgebraMapKernel(phi) : Map -> MPol

 
Finite Dimensional Affine Algebras
      HasFiniteDimension(Q) : RngMPolRes -> BoolElt
      Dimension(Q) : RngMPolRes -> RngIntElt
      VectorSpace(Q) : RngMPolRes -> ModTupFld, Map
      MonomialBasis(Q) : RngMPolRes -> [ RngMPolResElt ]
      MatrixAlgebra(Q) : RngMPolRes -> AlgMat, Map
      RepresentationMatrix(f) : RngMPolResElt -> AlgMatElt
      IsUnit(f) : RngMPolResElt -> BoolElt
      IsNilpotent(f) : RngMPolResElt -> BoolElt, RngIntElt
      MinimalPolynomial(f) : RngMPolResElt -> RngUPol
      Example AlgAff_MinimalPolynomial (H108E3)

 
Affine Algebras which are Fields
      Example AlgAff_EllipticCurve (H108E4)
      Example AlgAff_Factorization (H108E5)
      Example AlgAff_MultiExtension (H108E6)

 
Rings and Fields of Fractions of Affine Algebras
      RingOfFractions(Q) : RngMPolRes -> RngFunFrac
      Numerator(a) : RngFunFracElt -> RngMPolResElt
      Example AlgAff_FieldOfFractions (H108E7)
      Example AlgAff_Extension (H108E8)

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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013