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<head><title>minimalPresentation(Ring) -- compute a minimal presentation of a quotient ring</title>
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<div><h1>minimalPresentation(Ring) -- compute a minimal presentation of a quotient ring</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>S = minimalPresentation R</tt><br/><tt>S = prune R</tt></div>
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<li><span>Function: <a href="_minimal__Presentation.html" title="compute a minimal presentation">minimalPresentation</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>R</tt>, <span>a <a href="___Ring.html">ring</a></span>, a quotient ring</span></li>
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<li><div class="single">Consequences:<ul><li>the isomorphism from <tt>R</tt> to <tt>S</tt> is stored as <tt>R.minimalPresentationMap</tt> and the inverse of this map is stored as <tt>R.minimalPresentationMapInv</tt></li>
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<li><div class="single">Outputs:<ul><li><span><tt>S</tt>, <span>a <a href="___Ring.html">ring</a></span>, a quotient ring, minimally presented if <tt>R</tt> is homogeneous, isomorphic to <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>Exclude => ...</tt> (missing documentation<!-- tag: minimalPresentation(..., Exclude => ...) -->), </span></li>
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<div class="single"><h2>Description</h2>
<div>The computation is accomplished by considering the relations of <tt>R</tt>. If a variable occurs as a term of a relation of <tt>R</tt> and in no other terms of the same polynomial, then the variable is replaced by the remaining terms and removed from the ring. A minimal generating set for the resulting defining ideal is then computed and the new quotient ring is returned. If <tt>R</tt> is not homogeneous, then an attempt is made to improve the presentation.<table class="examples"><tr><td><pre>i1 : R = ZZ/101[x,y,z,u,w]/ideal(x-x^2-y,z+x*y,w^2-u^2);</pre>
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<tr><td><pre>i2 : minimalPresentation(R)
ZZ
---[x, u, w]
101
o2 = ------------
2 2
- u + w
o2 : QuotientRing</pre>
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<tr><td><pre>i3 : R.minimalPresentationMap
ZZ
---[x, u, w]
101 2 3 2
o3 = map(------------,R,{x, - x + x, x - x , u, w})
2 2
- u + w
ZZ
---[x, u, w]
101
o3 : RingMap ------------ <--- R
2 2
- u + w</pre>
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<tr><td><pre>i4 : R.minimalPresentationMapInv
ZZ
---[x, u, w]
101
o4 = map(R,------------,{x, u, w})
2 2
- u + w
ZZ
---[x, u, w]
101
o4 : RingMap R <--- ------------
2 2
- u + w</pre>
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If the Exclude option is present, then those variables with the given indices are not simplified away (remember that ring variable indices start at 0).<table class="examples"><tr><td><pre>i5 : R = ZZ/101[x,y,z,u,w]/ideal(x-x^2-y,z+x*y,w^2-u^2);</pre>
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<tr><td><pre>i6 : minimalPresentation(R, Exclude=>{1})
ZZ
---[x, y, u, w]
101
o6 = -------------------------
2 2 2
(- x + x - y, - u + w )
o6 : QuotientRing</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_minimal__Presentation_lp__Ideal_rp.html" title="compute a minimal presentation of the quotient ring defined by an ideal">minimalPresentation(Ideal)</a> -- compute a minimal presentation of the quotient ring defined by an ideal</span></li>
<li><span><a href="_minimal__Presentation_lp__Ideal_rp.html" title="compute a minimal presentation of the quotient ring defined by an ideal">prune(Ideal)</a> -- compute a minimal presentation of the quotient ring defined by an ideal</span></li>
<li><span><a href="_trim_lp__Ring_rp.html" title="">trim(Ring)</a></span></li>
<li><span><a href="_trim_lp__Quotient__Ring_rp.html" title="">trim(QuotientRing)</a></span></li>
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