In Macaulay2, each ideal comes equipped with a matrix of generators. It is the number of columns of this matrix which is returned. If the ideal is homogeneous, this may or may not be the number of minimal generators.
i1 : R = QQ[a..d];
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i2 : I = ideal(a^2-b*d, a^2-b*d, c^2, d^2);
o2 : Ideal of R
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i3 : numgens I
o3 = 4
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In order to find a more efficient set of of generators, use
mingens or
trim.
i4 : mingens I
o4 = | d2 c2 a2-bd |
1 3
o4 : Matrix R <--- R
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i5 : numgens trim I
o5 = 3
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