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<head><title>regularSequence -- generates a regular sequence</title>
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<div><h1>regularSequence -- generates a regular sequence</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>regularSequence(I,A)</tt></div>
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<li><div class="single">Inputs:<ul><li><span><tt>I</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span></span></li>
<li><span><tt>A</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring.html">ring</a></span></span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span></span></li>
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<li><div class="single"><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><tt>Attempts => </tt><span><span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, <span>default value 100</span>, number of attempts made to generate a regular sequence</span></span></li>
<li><span><tt>Bound => </tt><span><span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, <span>default value 1</span>, bound on the value of the random coefficients</span></span></li>
<li><span><tt>Sparseness => </tt><span><span>a <a href="../../Macaulay2Doc/html/___R__R.html">real number</a></span>, <span>default value .5</span>, between 0 and 1 giving the frequency of the coefficients being equal to zero</span></span></li>
<li><span><tt>Maximal => </tt><span><span>a <a href="../../Macaulay2Doc/html/___Boolean.html">Boolean value</a></span>, <span>default value true</span>, whether to insist on searching for a maximal regular sequence</span></span></li>
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<div class="single"><h2>Description</h2>
<div>Given a ring and an ideal, <tt>regularSequence</tt> attempts to generate a regular sequence contained in <tt>I</tt>. The algorithm is based on one found in Chapter 5.5 of W. Vasconcelos' book: <em>Computational Methods in Commutative Algebra and Algebraic Geometry</em>.<table class="examples"><tr><td><pre>i1 : A = ZZ/5051[x, y, z];</pre>
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<tr><td><pre>i2 : I = ideal (x, x*y, y*z);
o2 : Ideal of A</pre>
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<tr><td><pre>i3 : X = regularSequence(I,A)
o3 = | xy-x -xy+yz |
1 2
o3 : Matrix A <--- A</pre>
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<tr><td><pre>i4 : isRegularSequence(X,A)
o4 = true</pre>
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Here are examples with optional inputs:<table class="examples"><tr><td><pre>i5 : A = ZZ/5051[x, y, z];</pre>
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<tr><td><pre>i6 : I = ideal (x, x*y, y*z);
o6 : Ideal of A</pre>
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<tr><td><pre>i7 : regularSequence(I,A,Attempts=>1,Bound=>100,Sparseness=>.9)
--warning: no maximal regular sequence found
o7 = 0
1
o7 : Matrix A <--- 0</pre>
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Here are examples with the optional input <tt>Maximal => false</tt>:<table class="examples"><tr><td><pre>i8 : x = symbol x; y = symbol y;</pre>
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<tr><td><pre>i10 : n = 2;</pre>
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<tr><td><pre>i11 : A = ZZ/101[x_(1,1)..x_(n,n),y_(1,1)..y_(n,n)];</pre>
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<tr><td><pre>i12 : X = transpose genericMatrix(A,n,n);
2 2
o12 : Matrix A <--- A</pre>
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<tr><td><pre>i13 : Y = transpose genericMatrix(A,y_(1,1),n,n);
2 2
o13 : Matrix A <--- A</pre>
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<tr><td><pre>i14 : b = ideal(X*Y - Y*X);
o14 : Ideal of A</pre>
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<tr><td><pre>i15 : B = A/b;</pre>
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<tr><td><pre>i16 : regularSequence(B,Attempts=>1,Maximal=>false)
o16 = | -y_(2,1) -x_(2,2)-y_(1,1) x_(1,2)+y_(1,1)
-----------------------------------------------------------------------
x_(1,2)-x_(2,1)-x_(2,2)-y_(1,2)-y_(2,2) x_(1,1)
-----------------------------------------------------------------------
-x_(2,2)-y_(1,1)+y_(1,2)-y_(2,2) |
1 6
o16 : Matrix B <--- B</pre>
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<p>This symbol is provided by the package <a href="index.html" title="computations involving regular sequences">Depth</a>.</p>
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<div class="waystouse"><h2>Ways to use <tt>regularSequence</tt> :</h2>
<ul><li>regularSequence(Ideal,Ring)</li>
<li>regularSequence(Ring)</li>
</ul>
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