<?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>standardPairs -- find the standard pairs of a monomial ideal</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="_stash__Value.html">next</a> | <a href="_standard__Form.html">previous</a> | <a href="_stash__Value.html">forward</a> | <a href="_standard__Form.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>standardPairs -- find the standard pairs of a monomial 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>standardPairs I</tt></div>
</dd></dl>
</div>
</li>
<li><div class="single">Inputs:<ul><li><span><tt>I</tt>, <span>a <a href="___Monomial__Ideal.html">monomial ideal</a></span></span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span>a <a href="___List.html" title="the class of all lists -- {...}">List</a> of standard pairs for <tt>I</tt></span></li>
</ul>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div>The standard monomials of a monomial ideal <tt>I</tt> (those monomials that are not in <tt>I</tt>) can be enumerated as follows. Given a monomial <tt>m</tt> and a subset <tt>F</tt> of the variables, the pair <tt>(m,F)</tt> indexes the set of monomials of the form <tt>mm'</tt> where the monomial <tt>m'</tt> is supported on <tt>F</tt>. A <a href="___List.html" title="the class of all lists -- {...}">List</a> of pairs <tt>(m, F)</tt> form <em>standard pairs</em> for the monomial ideal <tt>I</tt> if it satisfies the following three conditions:<ul><li>for each pair, the monomial <tt>m</tt> is supported on the complement of <tt>F</tt>;</li>
<li>all of the monomials represented by a pair are standard;</li>
<li>the pairs index disjoint sets of monomials.</li>
</ul>
<table class="examples"><tr><td><pre>i1 : R = QQ[x,y,z];</pre>
</td></tr>
<tr><td><pre>i2 : I = monomialIdeal(x*y^3*z, x*y^2*z^2, y^3*z^2, y^2*z^3)
3 2 2 3 2 2 3
o2 = monomialIdeal (x*y z, x*y z , y z , y z )
o2 : MonomialIdeal of R</pre>
</td></tr>
<tr><td><pre>i3 : standardPairs I
2 2 2
o3 = {{1, {x, z}}, {y, {x, z}}, {1, {x, y}}, {z, {y}}, {y z, {x}}, {y z ,
------------------------------------------------------------------------
{}}}
o3 : List</pre>
</td></tr>
</table>
<p/>
The standard pairs are computed with Algorithm 3.2.5 in Gröbner Deformations of Hypergeometric Differential Equations, by Mutsumi Saito, Bernd Sturmfels and Nobuki Takayama; Algorithms and Computation in Mathematics 6, Springer-Verlag, 2000. Implemented by Gregory G. Smith.<p/>
For more information, see the <em>Monomial ideals</em> chapter in Computations in algebraic geometry with Macaulay2, edited by David Eisenbud, Daniel R. Grayson, Michael E. Stillman, and Bernd Sturmfels, Algorithms and Computations in Mathematics 8, Springer-Verlag, 2001.</div>
</div>
<div class="waystouse"><h2>Ways to use <tt>standardPairs</tt> :</h2>
<ul><li>standardPairs(MonomialIdeal)</li>
<li>standardPairs(MonomialIdeal,List)</li>
</ul>
</div>
</div>
</body>
</html>
