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

Synopsis

• Usage:

  B = basisElements J

  B = basisElements F

• Inputs:
  • J, an instance of the type InvolutiveBasis
  • F, an instance of the type FactorModuleBasis
• Outputs:
  • B, a matrix

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 {y, z}, set {z}, set {z}, set {x}}

o10 : List

See also

• multVar -- extract the sets of multiplicative variables for each generator (in several contexts)
• janetBasis -- compute Janet basis for an ideal or a submodule of a free module
• factorModuleBasis -- enumerate standard monomials