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<head><title>syz(GroebnerBasis) -- retrieve the syzygy matrix</title>
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<div><h1>syz(GroebnerBasis) -- retrieve the syzygy 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>syz G</tt></div>
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<li><span>Function: <a href="_syz.html" title="the syzygy matrix">syz</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>, the Gröbner basis of a matrix <tt>h</tt></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Matrix.html">matrix</a></span>, the matrix of syzygies among the columns of <tt>h</tt></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="_syz_lp__Matrix_rp.html">Algorithm => ...</a>, -- compute the syzygy matrix</span></li>
<li><span><a href="_syz_lp__Matrix_rp.html">BasisElementLimit => ...</a>, -- compute the syzygy matrix</span></li>
<li><span><a href="_syz_lp__Matrix_rp.html">DegreeLimit => ...</a>, -- compute the syzygy matrix</span></li>
<li><span><a href="_syz_lp__Matrix_rp.html">GBDegrees => ...</a>, -- compute the syzygy matrix</span></li>
<li><span><a href="_syz_lp__Matrix_rp.html">HardDegreeLimit => ...</a>, -- compute the syzygy matrix</span></li>
<li><span><a href="_syz_lp__Matrix_rp.html">MaxReductionCount => ...</a>, -- compute the syzygy matrix</span></li>
<li><span><a href="_syz_lp__Matrix_rp.html">PairLimit => ...</a>, -- compute the syzygy matrix</span></li>
<li><span><a href="_syz_lp__Matrix_rp.html">StopBeforeComputation => ...</a>, -- compute the syzygy matrix</span></li>
<li><span><a href="_syz_lp__Matrix_rp.html">Strategy => ...</a>, -- compute the syzygy matrix</span></li>
<li><span><a href="_syz_lp__Matrix_rp.html">SyzygyLimit => ...</a>, -- compute the syzygy matrix</span></li>
<li><span><a href="_syz_lp__Matrix_rp.html">SyzygyRows => ...</a>, -- compute the syzygy matrix</span></li>
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<div class="single"><h2>Description</h2>
<div><p>Warning: the result may be zero if syzygies were not to be retained during the calculation, or if the computation was not continued to a high enough degree.</p>
<p>The matrix of syzygies is returned without removing non-minimal syzygies.</p>
<table class="examples"><tr><td><pre>i1 : R = QQ[a..g];</pre>
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<tr><td><pre>i2 : I = ideal"ab2-c3,abc-def,ade-bfg"
2 3
o2 = ideal (a*b - c , a*b*c - d*e*f, a*d*e - b*f*g)
o2 : Ideal of R</pre>
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<tr><td><pre>i3 : G = gb(I, Syzygies=>true);</pre>
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<tr><td><pre>i4 : syz G
o4 = {3} | -abc+def 0 -ade+bfg -d2e2f+b2cfg |
{3} | ab2-c3 -ade+bfg 0 c3de-b3fg |
{3} | 0 abc-def ab2-c3 -bc4+b2def |
3 4
o4 : Matrix R <--- R</pre>
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There appear to be 4 syzygies, but the last one is a combination of the first three:<table class="examples"><tr><td><pre>i5 : syz gens I
o5 = {3} | -abc+def 0 -ade+bfg |
{3} | ab2-c3 -ade+bfg 0 |
{3} | 0 abc-def ab2-c3 |
3 3
o5 : Matrix R <--- R</pre>
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<tr><td><pre>i6 : mingens image syz G
o6 = {3} | -abc+def 0 -ade+bfg |
{3} | ab2-c3 -ade+bfg 0 |
{3} | 0 abc-def ab2-c3 |
3 3
o6 : Matrix R <--- R</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_gb.html" title="compute a Gröbner basis">gb</a> -- compute a Gröbner basis</span></li>
<li><span><a href="_mingens.html" title="minimal generator matrix">mingens</a> -- minimal generator matrix</span></li>
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