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<head><title>degree(Ideal)</title>
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<div><h1>degree(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>degree I</tt></div>
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<li><span>Function: <a href="_degree.html" title="">degree</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>I</tt>, <span>an <a href="___Ideal.html">ideal</a></span>, in a polynomial ring or quotient of a polynomial ring</span></li>
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<li><div class="single">Outputs:<ul><li><span><span>an <a href="___Z__Z.html">integer</a></span>, the degree of the zero set of <tt>I</tt></span></li>
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<div class="single"><h2>Description</h2>
<div>The degree of an ideal <tt>I</tt> in a ring <tt>S</tt> is the degree of the module <tt>S/I</tt>. See <a href="_degree_lp__Module_rp.html" title="">degree(Module)</a> for more details.<table class="examples"><tr><td><pre>i1 : S = QQ[a..f];</pre>
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<tr><td><pre>i2 : I = ideal(a^5, b^5, c^5, d^5, e^5);
o2 : Ideal of S</pre>
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<tr><td><pre>i3 : degree I
o3 = 3125</pre>
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<tr><td><pre>i4 : degree(S^1/I)
o4 = 3125</pre>
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If the ideal is not homogeneous, then the degree returned is the degree of the ideal of initial monomials (which is homogeneous). If the monomial order is a degree order (the default), this is the same as the degree of the projective closure of the zero set of <tt>I</tt>.<table class="examples"><tr><td><pre>i5 : I = intersect(ideal(a-1,b-1,c-1),ideal(a-2,b-1,c+1),ideal(a-4,b+7,c-3/4));
o5 : Ideal of S</pre>
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<tr><td><pre>i6 : degree I
o6 = 3</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_dim.html" title="compute the Krull dimension">dim</a> -- compute the Krull dimension</span></li>
<li><span><a href="_codim.html" title="compute the codimension">codim</a> -- compute the codimension</span></li>
<li><span><a href="_genus.html" title="arithmetic genus">genus</a> -- arithmetic genus</span></li>
<li><span><a href="_genera.html" title="list of the successive linear sectional arithmetic genera">genera</a> -- list of the successive linear sectional arithmetic genera</span></li>
<li><span><a href="_hilbert__Series.html" title="compute the Hilbert series">hilbertSeries</a> -- compute the Hilbert series</span></li>
<li><span><a href="_reduce__Hilbert.html" title="reduce a Hilbert series expression">reduceHilbert</a> -- reduce a Hilbert series expression</span></li>
<li><span><a href="_poincare.html" title="assemble degrees into polynomial">poincare</a> -- assemble degrees into polynomial</span></li>
<li><span><a href="_hilbert__Polynomial.html" title="compute the Hilbert polynomial">hilbertPolynomial</a> -- compute the Hilbert polynomial</span></li>
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