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<head><title>isHomogeneous -- whether something is homogeneous (graded)</title>
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<div><h1>isHomogeneous -- whether something is homogeneous (graded)</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>isHomogeneous x</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>x</tt>, a <a href="___Ring.html" title="the class of all rings">Ring</a>, <a href="___Ring__Element.html" title="the class of all ring elements handled by the engine">RingElement</a>, <a href="___Vector.html" title="the class of all elements of free modules that are handled by the engine">Vector</a>, <a href="___Matrix.html" title="the class of all matrices">Matrix</a>, <a href="___Ideal.html" title="the class of all ideals">Ideal</a>, <a href="___Module.html" title="the class of all modules">Module</a>, <a href="___Ring__Map.html" title="the class of all ring maps">RingMap</a>, <a href="___Chain__Complex.html" title="the class of all chain complexes">ChainComplex</a>, or <a href="___Chain__Complex__Map.html" title="the class of all maps between chain complexes">ChainComplexMap</a></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Boolean.html">Boolean value</a></span>, whether <tt>x</tt> is homogeneous.</span></li>
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<div class="single"><h2>Description</h2>
<div><table class="examples"><tr><td><pre>i1 : isHomogeneous(ZZ)
o1 = true</pre>
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<tr><td><pre>i2 : isHomogeneous(ZZ[x])
o2 = true</pre>
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<tr><td><pre>i3 : isHomogeneous(ZZ[x]/(x^3-x-3))
o3 = false</pre>
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Rings may be graded, with generators having degree 0. For example, in the ring B below, every element of A has degree 0.<table class="examples"><tr><td><pre>i4 : A = QQ[a,b,c];</pre>
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<tr><td><pre>i5 : B = A[x,y];</pre>
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<tr><td><pre>i6 : isHomogeneous B
o6 = true</pre>
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<tr><td><pre>i7 : isHomogeneous ideal(a*x+y,y^3-b*x^2*y)
o7 = false</pre>
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Quotients of multigraded rings are homogeneous, if the ideal is also multigraded.<table class="examples"><tr><td><pre>i8 : R = QQ[a,b,c,Degrees=>{{1,1},{1,0},{0,1}}];</pre>
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<tr><td><pre>i9 : I = ideal(a-b*c);
o9 : Ideal of R</pre>
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<tr><td><pre>i10 : isHomogeneous I
o10 = true</pre>
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<tr><td><pre>i11 : isHomogeneous(R/I)
o11 = true</pre>
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<tr><td><pre>i12 : isHomogeneous(R/(a-b))
o12 = false</pre>
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A matrix is homogeneous if each entry is homogeneous of such a degree so that the matrix has a well-defined degree.<table class="examples"><tr><td><pre>i13 : S = QQ[a,b];</pre>
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<tr><td><pre>i14 : F = S^{-1,2}
2
o14 = S
o14 : S-module, free, degrees {1, -2}</pre>
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<tr><td><pre>i15 : isHomogeneous F
o15 = true</pre>
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<tr><td><pre>i16 : G = S^{1,2}
2
o16 = S
o16 : S-module, free, degrees {-1, -2}</pre>
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<tr><td><pre>i17 : phi = random(G,F)
o17 = {-1} | a2+5ab+2b2 0 |
{-2} | 2a3+8a2b+4ab2+8b3 8 |
2 2
o17 : Matrix S <--- S</pre>
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<tr><td><pre>i18 : isHomogeneous phi
o18 = true</pre>
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<tr><td><pre>i19 : degree phi
o19 = {0}
o19 : List</pre>
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Modules are homogeneous if their generator and relation matrices are homogeneous.<table class="examples"><tr><td><pre>i20 : M = coker phi
o20 = cokernel {-1} | a2+5ab+2b2 0 |
{-2} | 2a3+8a2b+4ab2+8b3 8 |
2
o20 : S-module, quotient of S</pre>
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<tr><td><pre>i21 : isHomogeneous(a*M)
o21 = true</pre>
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<tr><td><pre>i22 : isHomogeneous((a+1)*M)
o22 = false</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="_graded_spand_spmultigraded_sppolynomial_springs.html" title="">graded and multigraded polynomial rings</a></span></li>
</ul>
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<div class="waystouse"><h2>Ways to use <tt>isHomogeneous</tt> :</h2>
<ul><li>isHomogeneous(ChainComplex)</li>
<li>isHomogeneous(ChainComplexMap)</li>
<li>isHomogeneous(Ideal)</li>
<li>isHomogeneous(Matrix)</li>
<li>isHomogeneous(Module)</li>
<li>isHomogeneous(Ring)</li>
<li>isHomogeneous(RingElement)</li>
<li>isHomogeneous(RingMap)</li>
<li>isHomogeneous(Vector)</li>
</ul>
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