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<head><title>groebnerCone -- the cone whose interior weight vectors give the given initial ideal</title>
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<div><h1>groebnerCone -- the cone whose interior weight vectors give the given initial ideal</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>(C,H) = groebnerCone(inL,L)</tt></div>
</dd></dl>
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</li>
<li><div class="single">Inputs:<ul><li><span><tt>inL</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, of monomials which are to be the lead terms of the elements of <tt>L</tt></span></li>
<li><span><tt>L</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, of polynomials</span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><tt>C</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, the columns are the extremal rays of the Groebner cone</span></li>
<li><span><tt>H</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, the columns generate the largest linear space contained in the cone</span></li>
</ul>
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</li>
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<div class="single"><h2>Description</h2>
<div><table class="examples"><tr><td><pre>i1 : R = ZZ/32003[symbol a..symbol d]
o1 = R
o1 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i2 : inL = {c^4, b*d^2, b*c, b^2*d, b^3}
4 2 2 3
o2 = {c , b*d , b*c, b d, b }
o2 : List</pre>
</td></tr>
<tr><td><pre>i3 : L = {c^4-a*d^3, -c^3+b*d^2, b*c-a*d, -a*c^2+b^2*d, b^3-a^2*c}
4 3 3 2 2 2 3 2
o3 = {c - a*d , - c + b*d , b*c - a*d, - a*c + b d, b - a c}
o3 : List</pre>
</td></tr>
<tr><td><pre>i4 : weightVector(inL,L)
o4 = {8, 8, 3, 1}
o4 : List</pre>
</td></tr>
<tr><td><pre>i5 : groebnerCone(inL,L)
o5 = (| 0 0 |, | 1 0 |)
| 0 0 | | 0 1 |
| -2 -3 | | -2 3 |
| -3 -4 | | -3 4 |
o5 : Sequence</pre>
</td></tr>
<tr><td><pre>i6 : I = monomialCurveIdeal(R,{1,3,4})
3 2 2 2 3 2
o6 = ideal (b*c - a*d, c - b*d , a*c - b d, b - a c)
o6 : Ideal of R</pre>
</td></tr>
<tr><td><pre>i7 : time (inLs,Ls) = gfan I
LP algorithm being used: "cddgmp".
-- used 0.002484 seconds
2 2 2 3 2 2 3 2 2
o7 = ({{b*d , a*d, a*c , a c}, {c , a*d, a*c , a c}, {c , b*c, a*c , a c,
------------------------------------------------------------------------
3 3 4 2 2 3 3 2 3 2 3
a d}, {c , b*c, b , a*c , a c}, {c , b*c, b , a*c }, {c , b*c, b d, b },
------------------------------------------------------------------------
2 2 2 2 2 3 2 2 3 3
{b*d , b d, a*d, a c}, {b*d , b d, b , a*d}, {b*d , b*c, b d, b , a*d },
------------------------------------------------------------------------
4 2 2 3 3 2 2 2 3
{c , b*d , b*c, b d, b }}, {{- c + b*d , - b*c + a*d, a*c - b d, - b
------------------------------------------------------------------------
2 3 2 2 2 3 2 3 2
+ a c}, {c - b*d , - b*c + a*d, a*c - b d, - b + a c}, {c - b*d ,
------------------------------------------------------------------------
2 2 3 2 4 3 3 2
b*c - a*d, a*c - b d, - b + a c, - b + a d}, {c - b*d , b*c - a*d,
------------------------------------------------------------------------
4 3 2 2 3 2 3 2 3 2 2
b - a d, a*c - b d, - b + a c}, {c - b*d , b*c - a*d, b - a c, a*c
------------------------------------------------------------------------
2 3 2 2 2 3 2 3 2
- b d}, {c - b*d , b*c - a*d, - a*c + b d, b - a c}, {- c + b*d , -
------------------------------------------------------------------------
2 2 3 2 3 2 2 2 3
a*c + b d, - b*c + a*d, - b + a c}, {- c + b*d , - a*c + b d, b -
------------------------------------------------------------------------
2 3 2 2 2 3 2 4
a c, - b*c + a*d}, {- c + b*d , b*c - a*d, - a*c + b d, b - a c, - c
------------------------------------------------------------------------
3 4 3 3 2 2 2 3 2
+ a*d }, {c - a*d , - c + b*d , b*c - a*d, - a*c + b d, b - a c}})
o7 : Sequence</pre>
</td></tr>
<tr><td><pre>i8 : weightVector(inLs#0, Ls#0)
o8 = {1, 1, 4, 6}
o8 : List</pre>
</td></tr>
<tr><td><pre>i9 : scan(#inLs, i -> print weightVector(inLs#i, Ls#i));
{1, 1, 4, 6}
{1, 1, 4, 5}
{1, 1, 3, 2}
{2, 2, 3, 1}
{4, 4, 3, 1}
{6, 6, 3, 1}
{1, 1, 2, 4}
{2, 2, 1, 2}
{6, 6, 2, 1}
{8, 8, 3, 1}</pre>
</td></tr>
<tr><td><pre>i10 : scan(#inLs, i -> print groebnerCone(inLs#i, Ls#i));
(| 0 0 |, | 1 0 |)
| 0 0 | | 0 1 |
| 2 1 | | -2 3 |
| 3 2 | | -3 4 |
(| 0 0 |, | 1 0 |)
| 0 0 | | 0 1 |
| 2 1 | | -2 3 |
| 3 1 | | -3 4 |
(| 0 0 |, | 1 0 |)
| 0 0 | | 0 1 |
| 1 1 | | -2 3 |
| 0 1 | | -3 4 |
(| 0 0 |, | 1 0 |)
| 0 0 | | 0 1 |
| 1 0 | | -2 3 |
| 0 -1 | | -3 4 |
(| 0 0 |, | 1 0 |)
| 0 0 | | 0 1 |
| 0 -1 | | -2 3 |
| -1 -2 | | -3 4 |
(| 0 0 |, | 1 0 |)
| 0 0 | | 0 1 |
| -1 -2 | | -2 3 |
| -2 -3 | | -3 4 |
(| 0 0 |, | 1 0 |)
| 0 0 | | 0 1 |
| 1 0 | | -2 3 |
| 2 1 | | -3 4 |
(| 0 0 |, | 1 0 |)
| 0 0 | | 0 1 |
| 0 -1 | | -2 3 |
| 1 -1 | | -3 4 |
(| 0 0 |, | 1 0 |)
| 0 0 | | 0 1 |
| -1 -3 | | -2 3 |
| -1 -4 | | -3 4 |
(| 0 0 |, | 1 0 |)
| 0 0 | | 0 1 |
| -2 -3 | | -2 3 |
| -3 -4 | | -3 4 |</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_gfan.html" title="all initial ideals of an ideal">gfan</a> -- all initial ideals of an ideal</span></li>
<li><span><a href="_initial__Ideal.html" title="initial ideal with respect to a weight vector">initialIdeal</a> -- initial ideal with respect to a weight vector</span></li>
<li><span><a href="_weight__Vector.html" title="weight vector of a marked set of polynomials">weightVector</a> -- weight vector of a marked set of polynomials</span></li>
</ul>
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<div class="waystouse"><h2>Ways to use <tt>groebnerCone</tt> :</h2>
<ul><li>groebnerCone(List,List)</li>
</ul>
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