<?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>toDual -- inverse system</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="_to__External__String.html">next</a> | <a href="_to__C__C.html">previous</a> | <a href="_to__External__String.html">forward</a> | <a href="_to__C__C.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>toDual -- inverse system</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>g = toDual(d,f)</tt></div>
</dd></dl>
</div>
</li>
<li><div class="single">Inputs:<ul><li><span><tt>d</tt>, <span>an <a href="___Z__Z.html">integer</a></span></span></li>
<li><span><tt>f</tt>, <span>a <a href="___Matrix.html">matrix</a></span>, a homogeneous one row matrix over a polynomial ring R = k[x1 ... xn]</span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><tt>g</tt>, <span>a <a href="___Matrix.html">matrix</a></span>, a one row matrix</span></li>
</ul>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div>Returns generators for the intersection of the submodule I' = Hom(R/image f, E) and the submodule of E generated by y1^d ... yn^d. For this notation, and more details and examples, see <a href="_inverse_spsystems.html" title="">inverse systems</a>.<p/>
If I = ideal f contains the powers x1^(d+1), ..., xn^(d+1), then toDual(d,f) is a matrix whose entries correspond to the generators of Hom_R(R/image f, E).<table class="examples"><tr><td><pre>i1 : R = ZZ/32003[a..e];</pre>
</td></tr>
<tr><td><pre>i2 : f = matrix{{a^2, b^2, c^2, d^2, e^3, a*d-e^2}}
o2 = | a2 b2 c2 d2 e3 ad-e2 |
1 6
o2 : Matrix R <--- R</pre>
</td></tr>
<tr><td><pre>i3 : g = toDual(1,f)
o3 = {1} | abce |
{1} | bcde |
2 1
o3 : Matrix R <--- R</pre>
</td></tr>
<tr><td><pre>i4 : ideal fromDual g == ideal f
o4 = false</pre>
</td></tr>
<tr><td><pre>i5 : g = toDual(2,f)
o5 = {6} | abce |
{6} | bcde |
{6} | abcd+bce2 |
3 1
o5 : Matrix R <--- R</pre>
</td></tr>
<tr><td><pre>i6 : ideal fromDual g == ideal f
o6 = true</pre>
</td></tr>
<tr><td><pre>i7 : g = toDual(3,f)
o7 = {11} | abce |
{11} | bcde |
{11} | abcd+bce2 |
3 1
o7 : Matrix R <--- R</pre>
</td></tr>
<tr><td><pre>i8 : ideal fromDual g == ideal f
o8 = true</pre>
</td></tr>
</table>
</div>
</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="_from__Dual.html" title="ideal from inverse system">fromDual</a> -- ideal from inverse system</span></li>
</ul>
</div>
<div class="waystouse"><h2>Ways to use <tt>toDual</tt> :</h2>
<ul><li><span><tt>toDual(ZZ,Matrix)</tt> (missing documentation<!-- tag: (toDual,ZZ,Matrix) -->)</span></li>
</ul>
</div>
</div>
</body>
</html>
