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<head><title>factorModuleBasis -- enumerate standard monomials</title>
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<div><h1>factorModuleBasis -- enumerate standard monomials</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>F = factorModuleBasis(J)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>J</tt>, <span>an object of class <a href="___Involutive__Basis.html" title="the class of all involutive bases">InvolutiveBasis</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><tt>F</tt>, <span>an object of class <a href="___Factor__Module__Basis.html" title="the class of all factor module bases">FactorModuleBasis</a></span>, a partition of the set of monomials that are not leading monomial of any element of the module spanned by J, into monomial cones</span></li>
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<div class="single"><h2>Description</h2>
<div><p>The result represents a collection of finitely many cones of monomials, each cone being the set of multiples of a certain monomial by all monomials in certain variables; the generating monomials are accessed by <a href="_basis__Elements.html" title="extract the matrix of generators from an involutive basis or factor module basis">basisElements</a>; the sets of variables for each cone are obtained from <a href="_mult__Var.html" title="extract the sets of multiplicative variables for each generator (in several contexts)">multVar</a>.</p>
<table class="examples"><tr><td><pre>i1 : R = QQ[x,y,z];</pre>
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<tr><td><pre>i2 : M = matrix {{x*y,x^3*z}};
1 2
o2 : Matrix R <--- R</pre>
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<tr><td><pre>i3 : J = janetBasis M;</pre>
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<tr><td><pre>i4 : F = factorModuleBasis J
+--+------+
o4 = |1 |{z, y}|
+--+------+
|x |{z} |
+--+------+
| 2| |
|x |{z} |
+--+------+
| 3| |
|x |{x} |
+--+------+
o4 : FactorModuleBasis</pre>
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<tr><td><pre>i5 : basisElements F
o5 = | 1 x x2 x3 |
1 4
o5 : Matrix R <--- R</pre>
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<tr><td><pre>i6 : multVar F
o6 = {set {y, z}, set {z}, set {z}, set {x}}
o6 : List</pre>
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<table class="examples"><tr><td><pre>i7 : R = QQ[x,y];</pre>
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<tr><td><pre>i8 : M = matrix {{x*y-y^3, x*y^2, x*y-x}, {x, y^2, x}};
2 3
o8 : Matrix R <--- R</pre>
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<tr><td><pre>i9 : J = janetBasis M
+--------------+------+
o9 = || y3-x | |{y} |
|| 0 | | |
+--------------+------+
|| xy-x | |{y} |
|| x | | |
+--------------+------+
|| x2y-x2 | |{y} |
|| x2 | | |
+--------------+------+
|| x3 | |{y, x}|
|| x2 | | |
+--------------+------+
|| -x | |{y} |
|| xy-y2+x | | |
+--------------+------+
|| x2 | |{y} |
|| y3 | | |
+--------------+------+
|| -x2 ||{y} |
|| x2y-xy2+x2 || |
+--------------+------+
|| 0 | |{y, x}|
|| x3+2x2+y2 | | |
+--------------+------+
o9 : InvolutiveBasis</pre>
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<tr><td><pre>i10 : F = factorModuleBasis J
+------+--+
o10 = || 1 | |{}|
|| 0 | | |
+------+--+
|| y | |{}|
|| 0 | | |
+------+--+
|| y2 ||{}|
|| 0 || |
+------+--+
|| x | |{}|
|| 0 | | |
+------+--+
|| x2 ||{}|
|| 0 || |
+------+--+
|| 0 | |{}|
|| 1 | | |
+------+--+
|| 0 | |{}|
|| y | | |
+------+--+
|| 0 ||{}|
|| y2 || |
+------+--+
|| 0 | |{}|
|| x | | |
+------+--+
|| 0 ||{}|
|| x2 || |
+------+--+
o10 : FactorModuleBasis</pre>
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<tr><td><pre>i11 : basisElements F
o11 = | 1 y y2 x x2 0 0 0 0 0 |
| 0 0 0 0 0 1 y y2 x x2 |
2 10
o11 : Matrix R <--- R</pre>
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<tr><td><pre>i12 : multVar F
o12 = {set {}, set {}, set {}, set {}, set {}, set {}, set {}, set {}, set
-----------------------------------------------------------------------
{}, set {}}
o12 : List</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_janet__Basis.html" title="compute Janet basis for an ideal or a submodule of a free module">janetBasis</a> -- compute Janet basis for an ideal or a submodule of a free module</span></li>
<li><span><a href="_basis__Elements.html" title="extract the matrix of generators from an involutive basis or factor module basis">basisElements</a> -- extract the matrix of generators from an involutive basis or factor module basis</span></li>
<li><span><a href="_mult__Var.html" title="extract the sets of multiplicative variables for each generator (in several contexts)">multVar</a> -- extract the sets of multiplicative variables for each generator (in several contexts)</span></li>
</ul>
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<div class="waystouse"><h2>Ways to use <tt>factorModuleBasis</tt> :</h2>
<ul><li>factorModuleBasis(InvolutiveBasis)</li>
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