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Macaulay2Doc :: mingens(GroebnerBasis)

mingens(GroebnerBasis) -- (partially constructed) minimal generator matrix

Synopsis

Description

Every GroebnerBasis computation in Macaulay2 computes a generator matrix, in the process of constructing the Gröbner basis. If the original ideal or module is homogeneous, then the columns of this matrix form a minimal set of generators. In the inhomogeneous case, the columns generate, and an attempt is made to keep the size of the generating set small.

If the Gröbner basis is only partially constructed, the returned result will be a partial answer. In the graded case this set can be extended to a minimal set of generators for the ideal or module.

i1 : R = QQ[a..f]

o1 = R

o1 : PolynomialRing
i2 : M = genericSymmetricMatrix(R,a,3)

o2 = | a b c |
     | b d e |
     | c e f |

             3       3
o2 : Matrix R  <--- R
i3 : I = minors(2,M)

               2                                                  2         
o3 = ideal (- b  + a*d, - b*c + a*e, - c*d + b*e, - b*c + a*e, - c  + a*f, -
     ------------------------------------------------------------------------
                                             2
     c*e + b*f, - c*d + b*e, - c*e + b*f, - e  + d*f)

o3 : Ideal of R
i4 : G = gb(I, PairLimit=>5)

o4 = GroebnerBasis[status: PairLimit; all S-pairs handled up to degree 1]

o4 : GroebnerBasis
i5 : mingens G

o5 = | e2-df ce-bf cd-be |

             1       3
o5 : Matrix R  <--- R
i6 : mingens I

o6 = | e2-df ce-bf cd-be c2-af bc-ae b2-ad |

             1       6
o6 : Matrix R  <--- R

See also