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<div><a href="index.html" title="">PrimaryDecomposition</a> > <a href="_primary__Decomposition_lp..._cm_sp__Strategy_sp_eq_gt_sp..._rp.html" title="">primaryDecomposition(..., Strategy => ...)</a></div>
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<div><h1>primaryDecomposition(..., Strategy => ...)</h1>
<div class="single"><h2>Description</h2>
<div>The strategy option value should be one of the following.<ul><li>Monomial -- uses Alexander duality of a monomial ideal</li>
<li>Binomial -- finds a cellular resolution of a binomial ideal. NOT IMPLEMENTED YET.</li>
<li>EisenbudHunekeVasconcelos -- uses the algorithm of Eisenbud-Huneke-Vasconcelos</li>
<li>ShimoyamaYokoyama -- uses the algorithm of Shimoyama-Yokoyama</li>
<li>Hybrid -- uses parts of the above two algorithms</li>
<li>GTZ -- uses the algorithm of Gianni-Trager-Zacharias. NOT IMPLEMENTED YET.</li>
</ul>
The default strategy depends on the ideal. If the ideal is generated by monomials, then <tt>Strategy => Monomial</tt> is implied. In all other cases, the default is <tt>Strategy => ShimoyamaYokoyama</tt>.<h3>Strategy => Monomial</h3>
This strategy only works for monomial ideals, and is the default strategy for such ideals. See the chapter "Monomial Ideals" in the Macaulay2 book.<table class="examples"><tr><td><pre>i1 : Q = QQ[x,y]
o1 = Q
o1 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i2 : I = ideal(x^2,x*y)
2
o2 = ideal (x , x*y)
o2 : Ideal of Q</pre>
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<tr><td><pre>i3 : primaryDecomposition(I, Strategy => Monomial)
2
o3 = {ideal(x), ideal (x , y)}
o3 : List</pre>
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<h3>Strategy => EisenbudHunekeVasconcelos</h3>
See "Direct methods for primary decomposition" by Eisenbud, Huneke, and Vasconcelos, Invent. Math. 110, 207-235 (1992).<table class="examples"><tr><td><pre>i4 : Q = QQ[x,y]
o4 = Q
o4 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i5 : I = ideal(x^2,x*y)
2
o5 = ideal (x , x*y)
o5 : Ideal of Q</pre>
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<tr><td><pre>i6 : primaryDecomposition(I, Strategy => EisenbudHunekeVasconcelos)
2 2
o6 = {ideal(x), ideal (y , x*y, x )}
o6 : List</pre>
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<h3>Strategy => ShimoyamaYokoyama</h3>
This strategy is the default for non-monomial ideals. See "Localization and Primary Decomposition of Polynomial ideals" by Shimoyama and Yokoyama, J. Symb. Comp. 22, 247-277 (1996).<table class="examples"><tr><td><pre>i7 : Q = QQ[x,y]
o7 = Q
o7 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i8 : I = ideal(x^2,x*y)
2
o8 = ideal (x , x*y)
o8 : Ideal of Q</pre>
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<tr><td><pre>i9 : primaryDecomposition(I, Strategy => ShimoyamaYokoyama)
2
o9 = {ideal(x), ideal (y, x )}
o9 : List</pre>
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<h3>Strategy => Hybrid</h3>
Use a hybrid of the Eisenbud-Huneke-Vasconcelos and Shimoyama-Yokoyama strategies. The field <tt>Strategy</tt> is a list of two integers, indicating the strategy to use for finding associated primes and localizing, respectively. WARNING: Setting the second paramter to 1 works only if the ideal is homogeneous and equidimensional.<table class="examples"><tr><td><pre>i10 : Q = QQ[x,y]
o10 = Q
o10 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i11 : I = intersect(ideal(x^2), ideal(y^2))
2 2
o11 = ideal(x y )
o11 : Ideal of Q</pre>
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<tr><td><pre>i12 : primaryDecomposition(I, Strategy => new Hybrid from (1,1))
2 2
o12 = {ideal(x ), ideal(y )}
o12 : List</pre>
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<tr><td><pre>i13 : primaryDecomposition(I, Strategy => new Hybrid from (1,2))
2 2
o13 = {ideal(x ), ideal(y )}
o13 : List</pre>
</td></tr>
<tr><td><pre>i14 : primaryDecomposition(I, Strategy => new Hybrid from (2,1))
2 2
o14 = {ideal(x ), ideal(y )}
o14 : List</pre>
</td></tr>
<tr><td><pre>i15 : primaryDecomposition(I, Strategy => new Hybrid from (2,2))
2 2
o15 = {ideal(x ), ideal(y )}
o15 : List</pre>
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<h2>Further information</h2>
<ul><li><span>Default value: <a href="../../Macaulay2Doc/html/_null.html" title="the unique member of the empty class">null</a></span></li>
<li><span>Function: <span><a href="_primary__Decomposition.html" title="irredundant primary decomposition of an ideal">primaryDecomposition</a> -- irredundant primary decomposition of an ideal</span></span></li>
<li><span>Option name: <span><a href="../../Macaulay2Doc/html/___Strategy.html" title="name for an optional argument">Strategy</a> -- name for an optional argument</span></span></li>
</ul>
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