<?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>icFractions -- fractions integral over an affine domain</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="_ic__Map.html">next</a> | <a href="_ic__Frac__P_lp..._cm_sp__Verbosity_sp_eq_gt_sp..._rp.html">previous</a> | <a href="_ic__Map.html">forward</a> | <a href="_ic__Frac__P_lp..._cm_sp__Verbosity_sp_eq_gt_sp..._rp.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>icFractions -- fractions integral over an affine domain</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>icFractions R</tt></div>
</dd></dl>
</div>
</li>
<li><div class="single">Inputs:<ul><li><span><tt>R</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring.html">ring</a></span>, an affine domain</span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, a list of fractions over <tt>R</tt>, generating the integral closure of <tt>R</tt>, as an <tt>R</tt>-algebra.</span></li>
</ul>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div><div>If the integral closure of <tt>R</tt> has not yet been computed, that computation is performed first. No extra computation is then involved to find the fractions.</div>
<table class="examples"><tr><td><pre>i1 : R = QQ[x,y,z]/ideal(x^6-z^6-y^2*z^4);</pre>
</td></tr>
<tr><td><pre>i2 : icFractions R
3 2
x x
o2 = {--, --, x, y, z}
2 z
z
o2 : List</pre>
</td></tr>
<tr><td><pre>i3 : R' = integralClosure R
o3 = R'
o3 : QuotientRing</pre>
</td></tr>
<tr><td><pre>i4 : gens R'
o4 = {w , w , x, y, z}
4,0 3,0
o4 : List</pre>
</td></tr>
<tr><td><pre>i5 : netList (ideal R')_*
+--------------+
| 2 |
o5 = |w z - x |
| 3,0 |
+--------------+
|w z - w x |
| 4,0 3,0 |
+--------------+
| 2 |
|w x - w |
| 4,0 3,0 |
+--------------+
| 2 2 2|
|w - y - z |
| 4,0 |
+--------------+</pre>
</td></tr>
</table>
<div>Notice that the <i>i</i>-th fraction corresponds to the <i>i</i>-th generator of the integral closure. For instance, the variable <i>w<sub>(</sub>3,0) = x<sup>2</sup> /z</i>.</div>
</div>
</div>
<div class="single"><h2>Caveat</h2>
<div><div>(a) Currently in Macaulay2, fractions over quotients of polynomial rings do not have a nice normal form. In particular, sometimes the fractions are ’simplified’ to give much nastier looking fractions. We hope that in the near future, this misfeature will be corrected. (b) If you want to control the computation of the integral closure via optional arguments, then make sure you call <a href="_integral__Closure_lp__Ring_rp.html" title="compute the integral closure (normalization) of an affine domain">integralClosure(Ring)</a> first, since <tt>icFractions</tt> does not have optional arguments.</div>
</div>
</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="_integral__Closure.html" title="integral closure of an ideal or a domain">integralClosure</a> -- integral closure of an ideal or a domain</span></li>
<li><span><a href="_ic__Map.html" title="natural map from an affine domain into its integral closure">icMap</a> -- natural map from an affine domain into its integral closure</span></li>
</ul>
</div>
<div class="waystouse"><h2>Ways to use <tt>icFractions</tt> :</h2>
<ul><li>icFractions(Ring)</li>
</ul>
</div>
</div>
</body>
</html>
