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<div><h1>degree(Module)</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 M</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>M</tt>, <span>a <a href="___Module.html">module</a></span>, over a polynomial ring or quotient of a polynomial ring, over a field k</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 <tt>M</tt></span></li>
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<div class="single"><h2>Description</h2>
<div>We assume that <tt>M</tt> is a graded (homogeneous) module over a singly graded polynomal ring or a quotient of a polynomial ring, over a field <tt>k</tt>.<p>If <tt>M</tt> is finite dimensional over <tt>k</tt>, the degree of <tt>M</tt> is its dimension over <tt>k</tt>. Otherwise, the degree of <tt>M</tt> is the multiplicity of <tt>M</tt>, i.e., the integer <tt>d</tt> such that the Hilbert polynomial of <tt>M</tt> has the form <tt>z |--> d z^e/e! + lower terms in z.</tt></p>
<table class="examples"><tr><td><pre>i1 : R = ZZ/101[t,x,y,z];</pre>
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<tr><td><pre>i2 : degree (R^1 / (ideal vars R)^6)
o2 = 126</pre>
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<tr><td><pre>i3 : degree minors_2 matrix {{t,x,y},{x,y,z}}
o3 = 3</pre>
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<p>The algorithm computes the <a href="_hilbert__Series.html" title="compute the Hilbert series">hilbertSeries</a> of <tt>M</tt> (as a rational function), divides both numerator and denominator by <tt>1-T</tt> as often as possible, then evaluates both at <tt>T=1</tt> and returns the resulting quotient as a (possibly rational) number. When the module has finite length, then the rational function is a polynomial, and evaluating it at 1 returns the dimension over the ground field, which for a graded (homogenous) is the same as the length.</p>
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<div class="single"><h2>Caveat</h2>
<div>If the base ring is <a href="___Z__Z.html" title="the class of all integers">ZZ</a>, or the module is not homogeneous, it is likely that the answer is not what you would expect. Similarly, if the degrees of the variables are not all <tt>{1}</tt>, then the answer is harder to interpret. See <a href="_heft_spvectors.html" title="">heft vectors</a> and <a href="_multidegree.html" title="multidegree">multidegree</a>.</div>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_hilbert__Polynomial.html" title="compute the Hilbert polynomial">hilbertPolynomial</a> -- compute the Hilbert polynomial</span></li>
<li><span><a href="_is__Homogeneous.html" title="whether something is homogeneous (graded)">isHomogeneous</a> -- whether something is homogeneous (graded)</span></li>
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