<?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>what is a Groebner basis?</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="___Groebner_spbasis_spexamples_spand_spapplications.html">next</a> | <a href="___Gröbner_spbases.html">previous</a> | <a href="___Groebner_spbasis_spexamples_spand_spapplications.html">forward</a> | backward | <a href="___Gröbner_spbases.html">up</a> | <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>
<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="___Gröbner_spbases.html" title="">Gröbner bases</a> > <a href="_what_spis_spa_sp__Groebner_spbasis_qu.html" title="">what is a Groebner basis?</a></div>
<hr/>
<div><h1>what is a Groebner basis?</h1>
<div>A Groebner basis is a specific generating set of an ideal or submodule over a polynomial ring. It is not minimal in general, but has extremely nice properties; it is reasonably easy to extract information about the ideal or submodule from a Groebner basis. We first describe Groebner bases in the important special case of an ideal in a polynomial ring. We will then describe Groebner bases of submodules, and over more general rings.<p/>
Let <i>R = k[x<sub>1</sub>, ..., x<sub>n</sub>]</i> be a polynomial ring over a field k, and let <i>I ⊂ R</i> be an ideal. A <i>monomial order</i> on R is a total order, <i>></i>, on the monomials of R, which satisfies two conditions: (1) <i>m > 1</i>, for every monomial m ≠1, and (2) the order is multiplicative: <i>m > n</i> implies that <i>mp > np</i>, for all monomials <i>m</i>, <i>n</i>, <i>p</i>.<p/>
In Macaulay2, each ring has a monomial order (also called a term order) associated with it. The default monomial order is <tt>GRevLex</tt>. See <a href="_monomial_sporderings.html" title="">monomial orderings</a> for more information.<p/>
Given a term order, the leading term is the one whose monomial is greatest in this order.<table class="examples"><tr><td><pre>i1 : R = ZZ/1277[a..d]
o1 = R
o1 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i2 : F = 5*a^3 + d^2 + a*d + b*c + 1
3 2
o2 = 5a + b*c + a*d + d + 1
o2 : R</pre>
</td></tr>
<tr><td><pre>i3 : leadTerm F
3
o3 = 5a
o3 : R</pre>
</td></tr>
</table>
For an ideal <i>I ⊂R</i>, the initial ideal <i>in(I)</i> is the ideal generated by the leading terms of the elements in I. A Groebner basis for I is a set of generators for I whose leading terms generate <i>in(I)</i>.<p/>
<table class="examples"><tr><td><pre>i4 : R = ZZ/1277[x,y];</pre>
</td></tr>
<tr><td><pre>i5 : I = ideal(x^3 - 2*x*y, x^2*y - 2*y^2 + x);
o5 : Ideal of R</pre>
</td></tr>
<tr><td><pre>i6 : leadTerm I
o6 = | y2 xy x2 |
1 3
o6 : Matrix R <--- R</pre>
</td></tr>
<tr><td><pre>i7 : gens gb I
o7 = | y2+638x xy x2 |
1 3
o7 : Matrix R <--- R</pre>
</td></tr>
</table>
The above example also shows that the leading terms of any set of generators of I do not necessarily generate in(I).<p/>
A Groebner basis for an ideal depends on the monomial ordering used in the ring .<table class="examples"><tr><td><pre>i8 : R = ZZ/1277[x,y, MonomialOrder => Lex];</pre>
</td></tr>
<tr><td><pre>i9 : I = ideal(x^3 - 2*x*y, x^2*y - 2*y^2 + x);
o9 : Ideal of R</pre>
</td></tr>
<tr><td><pre>i10 : gens gb I
o10 = | y3 x-2y2 |
1 2
o10 : Matrix R <--- R</pre>
</td></tr>
</table>
</div>
<div class="single"><h2>See also</h2>
<ul><li><span><a href="_monomial_sporderings.html" title="">monomial orderings</a></span></li>
<li><span><a href="_lead__Term.html" title="get the greatest term">leadTerm</a> -- get the greatest term</span></li>
<li><span><a href="___Gröbner_spbases.html" title="">Gröbner bases</a></span></li>
<li><span><a href="_normal_spforms.html" title="">normal forms</a></span></li>
</ul>
</div>
</div>
</body>
</html>
