<?xml version="1.0" encoding="utf-8" ?> <!-- for emacs: -*- coding: utf-8 -*- -->
<!-- Apache may like this line in the file .htaccess: AddCharset utf-8 .html -->
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" >
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head><title>minimalPresentation(Ideal) -- compute a minimal presentation of the quotient ring defined by an ideal</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="_minimal__Presentation_lp__Matrix_rp.html">next</a> | <a href="_minimal__Presentation.html">previous</a> | <a href="_minimal__Presentation_lp__Matrix_rp.html">forward</a> | <a href="_minimal__Presentation.html">backward</a> | up | <a href="index.html">top</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> | <a href="http://www.math.uiuc.edu/Macaulay2/">Macaulay2 web site</a></div>
</td>
</tr>
</table>
<hr/>
<div><h1>minimalPresentation(Ideal) -- compute a minimal presentation of the quotient ring defined by an ideal</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 = minimalPresentation I</tt></div>
</dd></dl>
</div>
</li>
<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>I</tt>, <span>an <a href="___Ideal.html">ideal</a></span>, in a ring <tt>S</tt></span></li>
</ul>
</div>
</li>
<li><div class="single">Consequences:<ul><li><tt>minimalPresentation</tt> stores the isomorphism from <tt>S/I</tt> to <tt>Q/J</tt> as <tt>I.cache.minimalPresentationMap</tt> , where <tt>I</tt> is the input ideal and the inverse of this map as <tt>I.cache.minimalPresentationMapInv</tt></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><tt>J</tt>, <span>an <a href="___Ideal.html">ideal</a></span>, in a new ring <tt>Q</tt> with <tt>S/I</tt> isomorphic to <tt>Q/J</tt></span></li>
</ul>
</div>
</li>
<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>
</ul>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div>If the ideal <tt>I</tt> is homogeneous, then the ideal <tt>J</tt>, in a new ring <tt>Q</tt> is the defining ideal for a minimal presentation of the ring <tt>S/I</tt> where <tt>S</tt> is the ring of <tt>I</tt>. This is accomplished as follows. If a variable occurs as a term of a generator of <tt>I</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 ideal is then computed. If <tt>I</tt> is not homogeneous, then an attempt is made to improve the presentation of <tt>S/I</tt>.<table class="examples"><tr><td><pre>i1 : C = ZZ/101[x,y,z,u,w];</pre>
</td></tr>
<tr><td><pre>i2 : I = ideal(x-x^2-y,z+x*y,w^2-u^2);
o2 : Ideal of C</pre>
</td></tr>
<tr><td><pre>i3 : minPres I
2 2
o3 = ideal(- u + w )
ZZ
o3 : Ideal of ---[x, u, w]
101</pre>
</td></tr>
<tr><td><pre>i4 : I.cache.minimalPresentationMap
ZZ 2 3 2
o4 = map(---[x, u, w],C,{x, - x + x, x - x , u, w})
101
ZZ
o4 : RingMap ---[x, u, w] <--- C
101</pre>
</td></tr>
<tr><td><pre>i5 : I.cache.minimalPresentationMapInv
ZZ
o5 = map(C,---[x, u, w],{x, u, w})
101
ZZ
o5 : RingMap C <--- ---[x, u, w]
101</pre>
</td></tr>
</table>
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>i6 : R = ZZ/101[x,y,z,u,w];</pre>
</td></tr>
<tr><td><pre>i7 : I = ideal(x-x^2-y,z+x*y,w^2-u^2);
o7 : Ideal of R</pre>
</td></tr>
<tr><td><pre>i8 : minimalPresentation(I, Exclude=>{1})
2 2 2
o8 = ideal (- x + x - y, - u + w )
ZZ
o8 : Ideal of ---[x, y, u, w]
101</pre>
</td></tr>
</table>
</div>
</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="_minimal__Presentation_lp__Ring_rp.html" title="compute a minimal presentation of a quotient ring">minimalPresentation(Ring)</a> -- compute a minimal presentation of a quotient ring</span></li>
<li><span><a href="_trim_lp__Ideal_rp.html" title="">trim(Ideal)</a></span></li>
</ul>
</div>
</div>
</body>
</html>
