<?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>pseudoRemainder -- compute the pseudo-remainder</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="_push__Forward_lp__Ring__Map_cm__Module_rp.html">next</a> | <a href="_pseudocode.html">previous</a> | <a href="_push__Forward_lp__Ring__Map_cm__Module_rp.html">forward</a> | <a href="_pseudocode.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>pseudoRemainder -- compute the pseudo-remainder</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>pseudoRemainder(f,g)</tt></div>
</dd></dl>
</div>
</li>
<li><div class="single">Inputs:<ul><li><span><tt>f</tt>, <span>a <a href="___Ring__Element.html">ring element</a></span></span></li>
<li><span><tt>g</tt>, <span>a <a href="___Ring__Element.html">ring element</a></span>, in the same polynomial ring <tt>R</tt> as <tt>f</tt></span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Ring__Element.html">ring element</a></span>, the pseudo remainder of the polynomial <tt>f</tt> by the polynomial <tt>g</tt></span></li>
</ul>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div>Let <tt>x</tt> be the first variable of <tt>R</tt> appearing in <tt>g</tt>. Suppose that <tt>g</tt> has degree <tt>d</tt> in <tt>x</tt>, and that the coefficient of <tt>x^d</tt> in <tt>g</tt> (as an element of <tt>R</tt>, but not involving the variable <tt>x</tt>) is <tt>c</tt>. The pseudo remainder of <tt>f</tt> by <tt>g</tt> is the polynomial <tt>h</tt> of degree less than <tt>d</tt> in <tt>x</tt> such that <tt>c^(e-d+1) * f = q*g + h</tt>, where <tt>f</tt> has degree <tt>e</tt> in <tt>x</tt>.<table class="examples"><tr><td><pre>i1 : R = QQ[x,y];</pre>
</td></tr>
<tr><td><pre>i2 : f = x^4
4
o2 = x
o2 : R</pre>
</td></tr>
<tr><td><pre>i3 : g = x^2*y + 13*x^2*y^4 +x*y^2-3*x - 1
2 4 2 2
o3 = 13x y + x y + x*y - 3x - 1
o3 : R</pre>
</td></tr>
<tr><td><pre>i4 : (lg, cg) = topCoefficients g
2 4
o4 = (x , 13y + y)
o4 : Sequence</pre>
</td></tr>
<tr><td><pre>i5 : h = pseudoRemainder(f,g)
6 4 3 4 2 2
o5 = - 27x*y + 87x*y - 2x*y + 14y - 27x*y + 6x*y - 6y + 27x + y + 9
o5 : R</pre>
</td></tr>
<tr><td><pre>i6 : (cg^3 * f - h) % g
o6 = 0
o6 : R</pre>
</td></tr>
<tr><td><pre>i7 : q = (cg^3 * f - h) // g
2 8 2 5 6 4 2 2 3 4 2
o7 = 169x y + 26x y - 13x*y + 39x*y + x y - x*y + 14y + 3x*y - 6y + y
------------------------------------------------------------------------
+ 9
o7 : R</pre>
</td></tr>
<tr><td><pre>i8 : cg^3*f == h + q*g
o8 = true</pre>
</td></tr>
</table>
</div>
</div>
<div class="single"><h2>Caveat</h2>
<div>There is no pseudo-division implemented, and the only way to change the notion of what the top variable is, is to change to a ring where the variables are in a different order</div>
</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="_top__Coefficients.html" title="first variable and its coefficient of a polynomial or matrix">topCoefficients</a> -- first variable and its coefficient of a polynomial or matrix</span></li>
</ul>
</div>
<div class="waystouse"><h2>Ways to use <tt>pseudoRemainder</tt> :</h2>
<ul><li>pseudoRemainder(RingElement,RingElement)</li>
</ul>
</div>
</div>
</body>
</html>
