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<head><title>initialIdeal -- initial ideal with respect to a weight vector</title>
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<div><h1>initialIdeal -- initial ideal with respect to a weight vector</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>M = initialIdeal(w,I) or M = initialIdeal(I)</tt></div>
</dd></dl>
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<li><div class="single">Inputs:<ul><li><span><tt>w</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, a positive weight vector</span></li>
<li><span><tt>I</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, in a polynomial ring (not a quotient ring)</span></li>
</ul>
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</li>
<li><div class="single">Outputs:<ul><li><span><tt>M</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, the ideal of lead polynomials under this weight vector </span></li>
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<div class="single"><h2>Description</h2>
<div><div>The weight vector should be totally positive, even in the homogeneous case. The result may or may not be a monomial ideal. When a weight vector is not specified, this simply uses the current term order.</div>
<table class="examples"><tr><td><pre>i1 : R = ZZ/32003[symbol a..symbol d]
o1 = R
o1 : PolynomialRing</pre>
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<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>
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<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>
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<tr><td><pre>i4 : weightVector(inL,L)
o4 = {8, 8, 3, 1}
o4 : List</pre>
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<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>
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<tr><td><pre>i6 : initialIdeal({8,8,3,1},ideal L)
2 4 2 3
o6 = ideal (b*d , b*c, c , b d, b )
o6 : Ideal of R</pre>
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<tr><td><pre>i7 : initialIdeal({5,5,2,1},ideal L)
2 4 3 2 3
o7 = ideal (b*c, b*d , c - a*d , b d, b )
o7 : Ideal of R</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="_weight__Vector.html" title="weight vector of a marked set of polynomials">weightVector</a> -- weight vector of a marked set of polynomials</span></li>
<li><span><a href="_groebner__Cone.html" title="the cone whose interior weight vectors give the given initial ideal">groebnerCone</a> -- the cone whose interior weight vectors give the given initial ideal</span></li>
</ul>
</div>
<div class="waystouse"><h2>Ways to use <tt>initialIdeal</tt> :</h2>
<ul><li>initialIdeal(Ideal)</li>
<li>initialIdeal(List,Ideal)</li>
</ul>
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