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<head><title>janetBasis -- compute Janet basis for an ideal or a submodule of a free module</title>
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<div><h1>janetBasis -- compute Janet basis for an ideal or a submodule of a free module</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>J = janetBasis M or J = janetBasis(C,n)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>M</tt>, <span>an object of class <a href="___Involutive__Basis.html" title="the class of all involutive bases">InvolutiveBasis</a></span></span></li>
<li><span><tt>M</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span></span></li>
<li><span><tt>M</tt>, <span>a <a href="../../Macaulay2Doc/html/___Groebner__Basis.html">Groebner basis</a></span></span></li>
<li><span><tt>C</tt>, <span>a <a href="../../Macaulay2Doc/html/___Chain__Complex.html">chain complex</a></span></span></li>
<li><span><tt>n</tt>, <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span></span></li>
</ul>
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<li><div class="single">Outputs:<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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<div class="single"><h2>Description</h2>
<div><p>If the argument for janetBasis is <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span> or <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span> or <span>a <a href="../../Macaulay2Doc/html/___Groebner__Basis.html">Groebner basis</a></span>, then J is a Janet basis for (the module generated by) M.</p>
<p>If the arguments for janetBasis are <span>a <a href="../../Macaulay2Doc/html/___Chain__Complex.html">chain complex</a></span> and <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, where C is the result of either <a href="_janet__Resolution.html" title="construct a free resolution for a given ideal or module using Janet bases">janetResolution</a> or <a href="../../Macaulay2Doc/html/_resolution.html" title="projective resolution">resolution</a> called with the optional argument 'Strategy => Involutive', then J is the Janet basis extracted from the n-th differential of C.</p>
<table class="examples"><tr><td><pre>i1 : R = QQ[x,y];</pre>
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<tr><td><pre>i2 : I = ideal(x^3,y^2);
o2 : Ideal of R</pre>
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<tr><td><pre>i3 : J = janetBasis I;</pre>
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<tr><td><pre>i4 : basisElements J
o4 = | y2 xy2 x3 x2y2 |
1 4
o4 : Matrix R <--- R</pre>
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<tr><td><pre>i5 : multVar J
o5 = {set {y}, set {y}, set {x, y}, set {y}}
o5 : List</pre>
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<table class="examples"><tr><td><pre>i6 : R = QQ[x,y];</pre>
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<tr><td><pre>i7 : M = matrix {{x*y-y^3, x*y^2, x*y-x}, {x, y^2, x}};
2 3
o7 : Matrix R <--- R</pre>
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<tr><td><pre>i8 : J = janetBasis M;</pre>
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<tr><td><pre>i9 : basisElements J
o9 = | y3-x xy-x x2y-x2 x3 -x x2 -x2 0 |
| 0 x x2 x2 xy-y2+x y3 x2y-xy2+x2 x3+2x2+y2 |
2 8
o9 : Matrix R <--- R</pre>
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<tr><td><pre>i10 : multVar J
o10 = {set {y}, set {y}, set {y}, set {x, y}, set {y}, set {y}, set {y}, set
-----------------------------------------------------------------------
{x, y}}
o10 : List</pre>
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<table class="examples"><tr><td><pre>i11 : R = QQ[x,y,z];</pre>
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<tr><td><pre>i12 : I = ideal(x,y,z);
o12 : Ideal of R</pre>
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<tr><td><pre>i13 : C = res(I, Strategy => Involutive)
1 3 3 1
o13 = R <-- R <-- R <-- R <-- 0
0 1 2 3 4
o13 : ChainComplex</pre>
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<tr><td><pre>i14 : janetBasis(C, 2)
+----------+---------+
o14 = |{1} | -y ||{z, y, x}|
|{1} | x || |
|{1} | 0 || |
+----------+---------+
|{1} | -z ||{z, y, x}|
|{1} | 0 || |
|{1} | x || |
+----------+---------+
|{1} | 0 ||{z, y} |
|{1} | -z || |
|{1} | y || |
+----------+---------+
o14 : InvolutiveBasis</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_janet__Mult__Var.html" title="return table of multiplicative variables for given module elements as determined by Janet division">janetMultVar</a> -- return table of multiplicative variables for given module elements as determined by Janet division</span></li>
<li><span><a href="_pommaret__Mult__Var.html" title="return table of multiplicative variables for given module elements as determined by Pommaret division">pommaretMultVar</a> -- return table of multiplicative variables for given module elements as determined by Pommaret division</span></li>
<li><span><a href="_is__Pommaret__Basis.html" title="check whether or not a given Janet basis is also a Pommaret basis">isPommaretBasis</a> -- check whether or not a given Janet basis is also a Pommaret basis</span></li>
<li><span><a href="_inv__Reduce.html" title="compute normal form modulo involutive basis by involutive reduction">invReduce</a> -- compute normal form modulo involutive basis by involutive reduction</span></li>
<li><span><a href="_inv__Syzygies.html" title="compute involutive basis of syzygies">invSyzygies</a> -- compute involutive basis of syzygies</span></li>
<li><span><a href="_janet__Resolution.html" title="construct a free resolution for a given ideal or module using Janet bases">janetResolution</a> -- construct a free resolution for a given ideal or module using Janet bases</span></li>
</ul>
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<div class="waystouse"><h2>Ways to use <tt>janetBasis</tt> :</h2>
<ul><li>janetBasis(ChainComplex,ZZ)</li>
<li>janetBasis(GroebnerBasis)</li>
<li>janetBasis(Ideal)</li>
<li>janetBasis(Matrix)</li>
<li><span><tt>janetBasis(Module)</tt> (missing documentation<!-- tag: (janetBasis,Module) -->)</span></li>
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