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<head><title>kernel(RingMap) -- kernel of a ringmap</title>
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<div><h1>kernel(RingMap) -- kernel of a ringmap</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>kernel f</tt></div>
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<li><span>Function: <a href="_kernel.html" title="kernel of a ringmap, matrix, or chain complex">kernel</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>f</tt>, <span>a <a href="___Ring__Map.html">ring map</a></span>, <tt>R</tt> --> <tt>S</tt></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>an <a href="___Ideal.html">ideal</a></span>, an ideal of <tt>R</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><tt>SubringLimit => </tt><span><span>an <a href="___Z__Z.html">integer</a></span>, <span>default value infinity</span>, stop the computation after this many elements of the kernel have been found</span></span></li>
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<div class="single"><h2>Description</h2>
<div><table class="examples"><tr><td><pre>i1 : R = QQ[a..d];</pre>
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<tr><td><pre>i2 : S = QQ[s,t];</pre>
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<tr><td><pre>i3 : F = map(S,R,{s^3, s^2*t, s*t^2, t^3})
3 2 2 3
o3 = map(S,R,{s , s t, s*t , t })
o3 : RingMap S <--- R</pre>
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<tr><td><pre>i4 : ker F
2 2
o4 = ideal (c - b*d, b*c - a*d, b - a*c)
o4 : Ideal of R</pre>
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<tr><td><pre>i5 : G = map(S,R,{s^5, s^3*t^2-t, s*t-s, t^5})
5 3 2 5
o5 = map(S,R,{s , s t - t, s*t - s, t })
o5 : RingMap S <--- R</pre>
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<tr><td><pre>i6 : ker(G, SubringLimit=>1)
2 10 2 7 2 7 2 2 4 3 5 6 3 3
o6 = ideal(a c + 10a b*c - 5a c + 25a b c + 2a c - 5a*b*c - 5a*b c +
------------------------------------------------------------------------
6 5 2 3 3 2 2 3 4 2 2 3 2
5a*c - a*b + 10a b c - 15a b*c - 5a*b c - 5a*b + 25a b c - 5a c -
------------------------------------------------------------------------
5 4 3 2 2 2
c + a - 10a*b + 20a b*c - 10a*b + 5a c - 5a*b - a)
o6 : Ideal of R</pre>
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In the case when everything is homogeneous, Hilbert functions are used to speed up the computations.</div>
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<div class="single"><h2>Caveat</h2>
<div>It should be possible to interrupt the computation and restart it, but this has not yet been implemented.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_substitution_spand_spmaps_spbetween_springs.html" title="">substitution and maps between rings</a></span></li>
<li><span><a href="_elimination_spof_spvariables.html" title="">elimination of variables</a></span></li>
<li><span><a href="_monomial__Curve__Ideal.html" title="make the ideal of a monomial curve">monomialCurveIdeal</a> -- make the ideal of a monomial curve</span></li>
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