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<head><title>multidegree -- multidegree</title>
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<div><h1>multidegree -- multidegree</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>multidegree M</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>M</tt>, <span>a <a href="___Module.html">module</a></span>, <span>an <a href="___Ideal.html">ideal</a></span>, or <span>a <a href="___Ring.html">ring</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span>the multidegree of <tt>M</tt>. If <tt>M</tt> is an ideal, the corresponding quotient ring is used.</span></li>
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<div class="single"><h2>Description</h2>
<div><p>The multidegree is defined on page 165 of <em>Combinatorial Commutative Algebra</em>, by Miller and Sturmfels, on page 165. It is an element of the degrees ring of <tt>M</tt>. Our implementation agrees with their definition provided the heft vector of the ring has every entry equal to 1. See also <em>Gröbner geometry of Schubert polynomials</em>, by Allen Knutson and Ezra Miller.</p>
<table class="examples"><tr><td><pre>i1 : S = QQ[a..d, Degrees => {{2,-1},{1,0},{0,1},{-1,2}}];</pre>
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<tr><td><pre>i2 : heft S
o2 = {1, 1}
o2 : List</pre>
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<tr><td><pre>i3 : multidegree ideal (b^2,b*c,c^2)
o3 = 3T T
0 1
o3 : ZZ[T , T ]
0 1</pre>
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<tr><td><pre>i4 : multidegree ideal a
o4 = 2T - T
0 1
o4 : ZZ[T , T ]
0 1</pre>
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<tr><td><pre>i5 : multidegree ideal (a^2,a*b,b^2)
2
o5 = 6T - 3T T
0 0 1
o5 : ZZ[T , T ]
0 1</pre>
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<tr><td><pre>i6 : describe ring oo
o6 = ZZ[T , T , Degrees => {2:1}, MonomialOrder => {MonomialSize => 32}, DegreeRank => 1, Inverses => true, Global => false]
0 1 {Weights => {2:-1} }
{GroupLex => 2 }
{Position => Up }</pre>
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<div class="single"><h2>Caveat</h2>
<div>This implementation is provisional in the case where the heft vector does not have every entry equal to 1.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_heft_spvectors.html" title="">heft vectors</a></span></li>
<li><span><a href="_degrees__Ring.html" title="the ring of degrees">degreesRing</a> -- the ring of degrees</span></li>
</ul>
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<div class="waystouse"><h2>Ways to use <tt>multidegree</tt> :</h2>
<ul><li>multidegree(Ideal)</li>
<li>multidegree(Module)</li>
<li>multidegree(Ring)</li>
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