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<head><title>part(List,RingElement) -- sum of terms of a polynomial of a given degree(s)</title>
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<div><h1>part(List,RingElement) -- sum of terms of a polynomial of a given degree(s)</h1>
<div class="single"><h2>Synopsis</h2>
<ul><li><span>Function: <a href="_part.html" title="select terms of a polynomial by degree or weight">part</a></span></li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div><div><h2>Synopsis</h2>
<ul><li><div class="list"><dl class="element"><dt class="heading">Usage: </dt><dd class="value"><div><tt>part(d,F)</tt><br/><tt>part_d F</tt></div>
</dd></dl>
</div>
</li>
<li>Inputs:<ul><li><span><tt>d</tt>, of integers denoting a multidegree</span></li>
<li><span><tt>F</tt>, an element in a polynomial ring</span></li>
</ul>
</li>
<li>Outputs:<ul><li><span><span>a <a href="___Ring__Element.html">ring element</a></span>, the degree <tt>d</tt> part of the polynomial <tt>F</tt></span></li>
</ul>
</li>
</ul>
If the polynomial ring is singly graded (the default case), then d may be an integer denoting this degree.<table class="examples"><tr><td><pre>i1 : R = QQ[a..d]
o1 = R
o1 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i2 : f = (a^2-b-1)*(c^3-b*d-2)
2 3 3 2 3 2 2
o2 = a c - b*c - a b*d - c + b d - 2a + b*d + 2b + 2
o2 : R</pre>
</td></tr>
<tr><td><pre>i3 : part({3},f)
3 2
o3 = - c + b d
o3 : R</pre>
</td></tr>
</table>
Here is an alternate syntax.<table class="examples"><tr><td><pre>i4 : part_{3} f
3 2
o4 = - c + b d
o4 : R</pre>
</td></tr>
</table>
In multigraded rings, degrees are lists of integers.<table class="examples"><tr><td><pre>i5 : R = QQ[a..d,Degrees=>{{1,0},{0,1},{1,-1},{0,-1}}]
o5 = R
o5 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i6 : F = a^3 + (b*d+1)^2
2 2 3
o6 = b d + a + 2b*d + 1
o6 : R</pre>
</td></tr>
<tr><td><pre>i7 : part_{0,0} F
2 2
o7 = b d + 2b*d + 1
o7 : R</pre>
</td></tr>
</table>
Polynomial rings over other polynomial rings are multigraded, by default.<table class="examples"><tr><td><pre>i8 : A = QQ[a,b,c]
o8 = A
o8 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i9 : B = A[x,y]
o9 = B
o9 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i10 : degree(a*x)
o10 = {1, 1}
o10 : List</pre>
</td></tr>
<tr><td><pre>i11 : part_{2,2} (a*x+b*y-1)^3
2 2 2 2
o11 = - 3a x - 6a*b*x*y - 3b y
o11 : B</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_degree.html" title="">degree</a></span></li>
<li><span><a href="_parts_lp__Ring__Element_rp.html" title="display a polynomial degree by degree">parts</a> -- display a polynomial degree by degree</span></li>
</ul>
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