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Macaulay2Doc :: Matrix % GroebnerBasis

Matrix % GroebnerBasis -- calculate the normal form of ring elements and matrices using a (partially computed) Gröbner basis

Synopsis

Description

In the following example, the seventh power of the trace of the matrix M is in the ideal generated by the entries of the cube of M. Since the ideal I is homogeneous, it is only required to compute the Gröbner basis in degrees at most seven.
i1 : R = QQ[a..i];
i2 : M = genericMatrix(R,a,3,3)

o2 = | a d g |
     | b e h |
     | c f i |

             3       3
o2 : Matrix R  <--- R
i3 : I = ideal(M^3);

o3 : Ideal of R
i4 : f = trace M

o4 = a + e + i

o4 : R
i5 : G = gb(I, DegreeLimit=>3)

o5 = GroebnerBasis[status: DegreeLimit; all S-pairs handled up to degree 3]

o5 : GroebnerBasis
i6 : f^7 % G == 0

o6 = false
i7 : gb(I, DegreeLimit=>7)	  	  

o7 = GroebnerBasis[status: DegreeLimit; all S-pairs handled up to degree 7]

o7 : GroebnerBasis
i8 : f^7 % G

o8 = 0

o8 : R
i9 : gb I

o9 = GroebnerBasis[status: done; S-pairs encountered up to degree 9]

o9 : GroebnerBasis
In these homogeneous situations, Macaulay2 only computes the Gröbner basis as far as required, as shown below.
i10 : I = ideal(M^3);

o10 : Ideal of R
i11 : G = gb(I, StopBeforeComputation=>true)

o11 = GroebnerBasis[status: not started; all S-pairs handled up to degree -1]

o11 : GroebnerBasis
i12 : f^7 % I

o12 = 0

o12 : R
i13 : status G

o13 = status: DegreeLimit; all S-pairs handled up to degree 7

See also