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<head><title>mingens(GroebnerBasis) -- (partially constructed) minimal generator matrix</title>
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<div><h1>mingens(GroebnerBasis) -- (partially constructed) minimal generator matrix</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>mingens G</tt></div>
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<li><span>Function: <a href="_mingens.html" title="minimal generator matrix">mingens</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>G</tt>, <span>a <a href="___Groebner__Basis.html">Groebner basis</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Matrix.html">matrix</a></span>, whose columns form a (partially computed) minimal generating set</span></li>
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<li><div class="single"><a href="_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><a href="___Complement.html">Strategy => ...</a>, </span></li>
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<div class="single"><h2>Description</h2>
<div>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.<p/>
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.<table class="examples"><tr><td><pre>i1 : R = QQ[a..f]
o1 = R
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : M = genericSymmetricMatrix(R,a,3)
o2 = | a b c |
| b d e |
| c e f |
3 3
o2 : Matrix R <--- R</pre>
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<tr><td><pre>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</pre>
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<tr><td><pre>i4 : G = gb(I, PairLimit=>5)
o4 = GroebnerBasis[status: PairLimit; all S-pairs handled up to degree 1]
o4 : GroebnerBasis</pre>
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<tr><td><pre>i5 : mingens G
o5 = | e2-df ce-bf cd-be |
1 3
o5 : Matrix R <--- R</pre>
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<tr><td><pre>i6 : mingens I
o6 = | e2-df ce-bf cd-be c2-af bc-ae b2-ad |
1 6
o6 : Matrix R <--- R</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="___Groebner__Basis.html" title="the class of all Gröbner bases">GroebnerBasis</a> -- the class of all Gröbner bases</span></li>
<li><span><a href="_gb.html" title="compute a Gröbner basis">gb</a> -- compute a Gröbner basis</span></li>
<li><span><a href="_generic__Symmetric__Matrix.html" title="make a generic symmetric matrix">genericSymmetricMatrix</a> -- make a generic symmetric matrix</span></li>
<li><span><a href="_minors_lp__Z__Z_cm__Matrix_rp.html" title="ideal generated by minors">minors</a> -- ideal generated by minors</span></li>
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