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<head><title>invReduce -- compute normal form modulo involutive basis by involutive reduction</title>
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<div><h1>invReduce -- compute normal form modulo involutive basis by involutive reduction</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>(r,c) = invReduce(p,J)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>p</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, the columns are to be reduced modulo J</span></li>
<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">Consequences:<ul><li>the columns of r are in normal form modulo J, and p = r + J#0 * c, where * is matrix multiplication</li>
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<li><div class="single">Outputs:<ul><li><span><tt>r</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, the normal form of (the columns of) <tt>p</tt> modulo <tt>J</tt></span></li>
<li><span><tt>c</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, the reduction coefficients</span></li>
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<div class="single"><h2>Description</h2>
<div><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+z, x*y+y*z+z*x, x*y*z-1}};
1 3
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 : p = matrix {{y,y^2,y^3}}
o4 = | y y2 y3 |
1 3
o4 : Matrix R <--- R</pre>
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<tr><td><pre>i5 : invReduce(p,J)
o5 = (| y -yz-z2 1 |, | 0 0 0 |)
| 0 1 y-z |
| 0 0 1 |
| 0 0 0 |
o5 : Sequence</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="_inv__Syzygies.html" title="compute involutive basis of syzygies">invSyzygies</a> -- compute involutive basis of syzygies</span></li>
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<div class="waystouse"><h2>Ways to use <tt>invReduce</tt> :</h2>
<ul><li>invReduce(Matrix,InvolutiveBasis)</li>
<li>invReduce(RingElement,InvolutiveBasis)</li>
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