<?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>leadTerm(Matrix) -- get the greatest term of each column</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="_lead__Term_lp__Ring__Element_rp.html">next</a> | <a href="_lead__Term_lp__Ideal_rp.html">previous</a> | <a href="_lead__Term_lp__Ring__Element_rp.html">forward</a> | <a href="_lead__Term_lp__Ideal_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>leadTerm(Matrix) -- get the greatest term of each column</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>leadTerm f</tt></div>
</dd></dl>
</div>
</li>
<li><span>Function: <a href="_lead__Term.html" title="get the greatest term">leadTerm</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>f</tt>, <span>a <a href="___Matrix.html">matrix</a></span>, in a polynomial ring</span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Matrix.html">matrix</a></span>, the lead term matrix of <tt>f</tt></span></li>
</ul>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div>In Macaulay2, each free module over a polynomial ring comes equipped with a <a href="_monomial_sporderings.html">monomial order</a> and this routine returns the matrix whose <tt>i</tt>-th column is the lead term of the <tt>i</tt> th column of <tt>f</tt>.<table class="examples"><tr><td><pre>i1 : R = QQ[a..d];</pre>
</td></tr>
<tr><td><pre>i2 : f = matrix{{0,a^2-b*c},{c,d}}
o2 = | 0 a2-bc |
| c d |
2 2
o2 : Matrix R <--- R</pre>
</td></tr>
<tr><td><pre>i3 : leadTerm f
o3 = | 0 a2 |
| c 0 |
2 2
o3 : Matrix R <--- R</pre>
</td></tr>
</table>
Coefficients are included in the result:<table class="examples"><tr><td><pre>i4 : R = ZZ[a..d][x,y,z];</pre>
</td></tr>
<tr><td><pre>i5 : f = matrix{{0,(a+b)*x^2},{c*x, (b+c)*y}}
o5 = | 0 x2a+x2b |
| xc yb+yc |
2 2
o5 : Matrix R <--- R</pre>
</td></tr>
<tr><td><pre>i6 : leadTerm f
o6 = | 0 x2a |
| xc 0 |
2 2
o6 : Matrix R <--- R</pre>
</td></tr>
</table>
The argument <tt>f</tt> can also be <span>a <a href="___Groebner__Basis.html">Groebner basis</a></span>, in which case the lead term matrix of the generating matrix of <tt>f</tt> is returned.</div>
</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="_lead__Coefficient.html" title="the leading coefficient">leadCoefficient</a> -- the leading coefficient</span></li>
<li><span><a href="_lead__Monomial.html" title="the leading monomial">leadMonomial</a> -- the leading monomial</span></li>
<li><span><a href="_lead__Component.html" title="the leading component of a vector or matrix">leadComponent</a> -- the leading component of a vector or matrix</span></li>
</ul>
</div>
</div>
</body>
</html>
