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basisElements -- extract the matrix of generators from an involutive basis or factor module basis

Description

If the argument of basisElements is an instance of the type InvolutiveBasis, then the columns of B are generators for the module spanned by the involutive basis. These columns form a Gr\"obner basis for this module.

If the argument of basisElements is an instance of the type FactorModuleBasis, then the columns of B are generators for the monomial cones in the factor module basis.

i1 : R = QQ[x,y];
 
i2 : I = ideal(x^3,y^2);

o2 : Ideal of R
 
i3 : J = janetBasis I;
 
i4 : basisElements J

o4 = | y2 xy2 x3 x2y2 |

             1      4
o4 : Matrix R  <-- R
 
i5 : R = QQ[x,y,z];
 
i6 : M = matrix {{x*y,x^3*z}};

             1      2
o6 : Matrix R  <-- R
 
i7 : J = janetBasis M;
 
i8 : F = factorModuleBasis J

     +--+------+
o8 = |1 |{z, y}|
     +--+------+
     |x |{z}   |
     +--+------+
     | 2|      |
     |x |{z}   |
     +--+------+
     | 3|      |
     |x |{x}   |
     +--+------+

o8 : FactorModuleBasis
 
i9 : basisElements F

o9 = | 1 x x2 x3 |

             1      4
o9 : Matrix R  <-- R
 
i10 : multVar F

o10 = {set {z, y}, set {z}, set {z}, set {x}}

o10 : List

See also

Ways to use basisElements:

  • basisElements(FactorModuleBasis)
  • basisElements(InvolutiveBasis)

For the programmer

The object basisElements is a method function.

The source of this document is in InvolutiveBases.m2:817:0.