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<head><title>points -- produces the ideal and initial ideal from the coordinates of a finite set of points</title>
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<div><h1>points -- produces the ideal and initial ideal from the coordinates of a finite set of points</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>(Q,inG,G) = points(M,R)</tt></div>
</dd></dl>
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</li>
<li><div class="single">Inputs:<ul><li><span><tt>M</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, in which each column consists of the coordinates of a point</span></li>
<li><span><tt>R</tt>, <span>a <a href="../../Macaulay2Doc/html/___Polynomial__Ring.html">polynomial ring</a></span>, coordinate ring of the affine space containing the points</span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><tt>Q</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, list of standard monomials</span></li>
<li><span><tt>inG</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, initial ideal of the set of points</span></li>
<li><span><tt>G</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, list of generators for Grobner basis for ideal of points</span></li>
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</li>
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<div class="single"><h2>Description</h2>
<div>This function uses the Buchberger-Moeller algorithm to compute a grobner basis for the ideal of a finite number of points in affine space. Here is a simple example.<table class="examples"><tr><td><pre>i1 : M = random(ZZ^3, ZZ^5)
o1 = | 8 9 9 3 9 |
| 3 0 1 2 7 |
| 6 9 4 9 9 |
3 5
o1 : Matrix ZZ <--- ZZ</pre>
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<tr><td><pre>i2 : R = QQ[x,y,z]
o2 = R
o2 : PolynomialRing</pre>
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<tr><td><pre>i3 : (Q,inG,G) = points(M,R)
2 2 2 3 2
o3 = ({1, z, y, x, z }, ideal (y*z, x*z, y , x*y, x , z ), {y*z - z - 9y +
------------------------------------------------------------------------
1 2 31 2 101 2 5 241 73
12z - 27, x*z + -z - 9x - --z + 99, y - ---z - -x - 7y + ---z - --,
2 2 90 3 18 5
------------------------------------------------------------------------
1 2 13 2 5 2 65 3 2
x*y - -z - 2x - 9y + --z + 12, x - -z - 12x + --z - 3, z - 19z +
6 6 6 6
------------------------------------------------------------------------
114z - 216})
o3 : Sequence</pre>
</td></tr>
<tr><td><pre>i4 : monomialIdeal G == inG
o4 = true</pre>
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<p/>
Next a larger example that shows that the Buchberger-Moeller algorithm in <tt>points</tt> may be faster than the alternative method using the intersection of the ideals for each point.<table class="examples"><tr><td><pre>i5 : R = ZZ/32003[vars(0..4), MonomialOrder=>Lex]
o5 = R
o5 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i6 : M = random(ZZ^5, ZZ^150)
o6 = | 0 1 0 7 0 3 1 8 2 3 0 0 7 5 0 3 0 5 6 6 9 3 5 2 5 1 1 3 3 6 0 6 1 9 5
| 6 8 2 9 3 0 8 9 6 8 8 3 5 7 5 4 3 3 6 6 3 0 4 2 3 1 3 3 4 1 2 9 7 3 3
| 7 0 2 3 3 4 6 0 8 0 2 7 7 7 0 8 1 3 8 9 9 8 0 7 3 5 7 7 9 6 3 3 2 8 5
| 2 9 8 5 9 0 7 1 0 0 7 5 4 8 8 6 6 5 5 8 3 3 4 0 7 6 9 5 5 3 6 9 8 0 9
| 4 4 6 3 3 7 5 3 7 8 9 9 4 5 2 1 5 2 9 6 1 5 9 5 5 1 0 8 2 4 0 2 9 1 3
------------------------------------------------------------------------
5 5 8 3 2 4 9 1 7 6 3 9 4 9 8 7 4 6 8 7 1 6 6 2 1 2 9 1 7 5 9 5 9 8 4 8
1 9 2 3 2 1 1 0 0 2 3 7 9 4 2 8 3 9 9 7 5 8 7 4 9 3 8 3 5 4 2 4 4 0 3 6
9 8 9 9 8 9 5 2 1 8 7 8 4 8 0 9 8 7 2 8 3 3 7 7 5 6 8 8 2 8 4 7 4 5 3 4
9 0 1 6 9 7 5 6 6 9 8 1 3 9 1 1 3 6 0 0 3 4 3 3 3 0 1 1 7 9 8 7 0 8 4 9
6 1 4 9 0 8 2 5 5 4 4 0 0 5 0 4 0 0 8 1 7 7 4 7 2 3 4 9 7 1 5 2 1 7 1 9
------------------------------------------------------------------------
7 8 4 9 0 7 1 6 1 4 1 3 1 6 6 6 3 5 6 6 3 1 3 1 1 3 5 0 9 8 9 6 4 8 1 1
8 8 0 8 4 4 5 5 6 3 5 8 9 2 8 7 9 8 6 8 0 2 7 8 1 0 0 6 7 6 1 5 2 4 6 6
8 4 1 4 6 4 2 3 7 8 9 3 0 4 9 5 7 4 4 7 8 5 1 2 6 4 5 2 5 9 5 1 2 9 5 2
2 0 7 2 8 7 8 5 0 6 3 7 9 9 5 4 5 0 5 8 6 9 6 4 0 6 4 5 7 9 9 6 9 1 6 0
1 5 6 9 3 5 5 6 1 5 8 9 3 7 9 3 7 6 2 6 8 6 0 6 3 1 3 3 1 2 6 7 6 0 1 9
------------------------------------------------------------------------
9 2 8 9 9 7 8 7 6 8 4 8 7 1 8 3 9 4 7 4 5 8 5 2 0 0 1 3 9 0 3 2 2 5 8 9
9 3 4 4 9 8 5 8 7 6 6 1 0 9 1 6 9 3 5 9 4 4 7 1 7 5 4 2 8 1 5 5 2 7 0 8
9 3 2 1 4 2 1 1 3 3 2 3 3 5 7 2 9 0 9 5 2 5 6 9 7 7 6 7 5 6 8 2 6 5 0 5
5 4 5 0 7 0 4 3 6 0 4 0 9 6 9 8 1 2 2 5 0 6 2 8 3 9 9 3 7 3 8 6 3 4 9 7
9 1 0 1 5 4 1 2 9 5 3 0 2 1 8 6 8 3 3 4 4 2 2 4 1 7 0 8 5 7 7 4 8 1 1 6
------------------------------------------------------------------------
2 6 7 5 2 9 3 |
5 1 9 5 0 0 8 |
6 0 6 0 1 9 5 |
6 9 0 9 3 2 2 |
2 5 0 4 5 0 6 |
5 150
o6 : Matrix ZZ <--- ZZ</pre>
</td></tr>
<tr><td><pre>i7 : time J = pointsByIntersection(M,R);
-- used 4.22736 seconds</pre>
</td></tr>
<tr><td><pre>i8 : time C = points(M,R);
-- used 0.535918 seconds</pre>
</td></tr>
<tr><td><pre>i9 : J == C_2
o9 = true</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_points__By__Intersection.html" title="computes ideal of point set by intersecting maximal ideals">pointsByIntersection</a> -- computes ideal of point set by intersecting maximal ideals</span></li>
</ul>
</div>
<div class="waystouse"><h2>Ways to use <tt>points</tt> :</h2>
<ul><li>points(Matrix,Ring)</li>
</ul>
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