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<head><title>mingens(Module) -- minimal generator matrix</title>
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<div><h1>mingens(Module) -- 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 I</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>I</tt>, <span>a <a href="___Module.html">module</a></span>, or an ideal</span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Matrix.html">matrix</a></span>, a matrix, whose columns generate I.</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>If I is homogeneous, then a matrix, whose columns minimally generate I, is returned.<table class="examples"><tr><td><pre>i1 : R = QQ[a..f]
o1 = R
o1 : PolynomialRing</pre>
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<tr><td><pre>i2 : I = ideal(a,b,c) * ideal(a,b,c)
2 2 2
o2 = ideal (a , a*b, a*c, a*b, b , b*c, a*c, b*c, c )
o2 : Ideal of R</pre>
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<tr><td><pre>i3 : mingens I
o3 = | c2 bc ac b2 ab a2 |
1 6
o3 : Matrix R <--- R</pre>
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If I is not homogeneous, then an attempt is made to find a more efficient generating matrix, one which is better than a Gröbner basis. There is no guarantee that the generating set is small, or that no subset also generates. The only thing known is that the entries do generate the ideal.<table class="examples"><tr><td><pre>i4 : J = ideal(a-1, b-2, c-3)
o4 = ideal (a - 1, b - 2, c - 3)
o4 : Ideal of R</pre>
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<tr><td><pre>i5 : I = J*J
2
o5 = ideal (a - 2a + 1, a*b - 2a - b + 2, a*c - 3a - c + 3, a*b - 2a - b +
------------------------------------------------------------------------
2
2, b - 4b + 4, b*c - 3b - 2c + 6, a*c - 3a - c + 3, b*c - 3b - 2c + 6,
------------------------------------------------------------------------
2
c - 6c + 9)
o5 : Ideal of R</pre>
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<tr><td><pre>i6 : mingens I
o6 = | c2-6c+9 bc-3b-2c+6 ac-3a-c+3 b2-4b+4 ab-2a-b+2 a2-2a+1 |
1 6
o6 : Matrix R <--- R</pre>
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Every module I in Macaulay2 is a submodule of a quotient of some ambient free module F. This routine returns a minimal, or improved generating set for the same module I. If you want to minimize the generators and the relations of a subquotient module, use <a href="_trim.html" title="minimize generators and relations">trim</a>. If you want a minimal presentation, then use <a href="_minimal__Presentation.html" title="compute a minimal presentation">minimalPresentation</a>.<table class="examples"><tr><td><pre>i7 : M = matrix{{a^2*b*c-d*e*f,a^3*c-d^2*f,a*d*f-b*c*e-1}}
o7 = | a2bc-def a3c-d2f -bce+adf-1 |
1 3
o7 : Matrix R <--- R</pre>
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<tr><td><pre>i8 : I = kernel M
o8 = image {4} | -bce+adf-1 0 a3c-d2f a4c-bcde-d -a3ce+d2ef a4df-bd2ef-a3 -a5df+abd2ef+a4 -a5cf+abcdef+adf |
{4} | 0 -bce+adf-1 -a2bc+def -a3bc+bce2+e a3df-de2f-a2 -a3bdf+bde2f+a2b a4bdf-abde2f-a3b a4bcf-abce2f-aef |
{3} | -a2bc+def -a3c+d2f 0 -a2bcd+a3ce -a5c+a2d2f -a2bd2f+a3def a3bd2f-a4def a3bcdf-a4cef |
3
o8 : R-module, submodule of R</pre>
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<tr><td><pre>i9 : J = image mingens I
o9 = image {4} | bce-adf+1 0 a3c-d2f |
{4} | 0 bce-adf+1 -a2bc+def |
{3} | a2bc-def a3c-d2f 0 |
3
o9 : R-module, submodule of R</pre>
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<tr><td><pre>i10 : I == J
o10 = true</pre>
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<tr><td><pre>i11 : trim I
o11 = image {4} | bce-adf+1 0 a3c-d2f |
{4} | 0 bce-adf+1 -a2bc+def |
{3} | a2bc-def a3c-d2f 0 |
3
o11 : R-module, submodule of R</pre>
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If the base ring is a polynomial ring (or quotient of one), then a Gröbner basis computation is started, and continued until all generators have been considered.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_trim.html" title="minimize generators and relations">trim</a> -- minimize generators and relations</span></li>
<li><span><a href="_minimal__Presentation.html" title="compute a minimal presentation">minimalPresentation</a> -- compute a minimal presentation</span></li>
<li><span><a href="_kernel.html" title="kernel of a ringmap, matrix, or chain complex">kernel</a> -- kernel of a ringmap, matrix, or chain complex</span></li>
<li><span><a href="_image.html" title="image of a map">image</a> -- image of a map</span></li>
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