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<head><title>elementary -- Elementary moves are used to reduce the target of a syzygy matrix</title>
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<div><h1>elementary -- Elementary moves are used to reduce the target of a syzygy matrix</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>g= elementary(f,k,m)</tt></div>
</dd></dl>
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<li><div class="single">Inputs:<ul><li><span><tt>f</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, whose target degrees are in ascending order</span></li>
<li><span><tt>k</tt>, <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, whose value is strictly less than the number of rows of f</span></li>
<li><span><tt>m</tt>, <span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, positive</span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><tt>g</tt>, <span>a <a href="../../Macaulay2Doc/html/___Matrix.html">matrix</a></span>, obtained from f by adding random multiples of the last row by polynomials in the first m variables to the k preceding rows, and then deleting the last row.</span></li>
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<div class="single"><h2>Description</h2>
<div><p>Factors out a general element, reducing the rank of f. More precisely, the routine adds random multiples of the last row, whose coefficients are polynomials in the first m variables, to the k preceding rows and drops the last row. For this to be effective, the target degrees of f must be in ascending order.</p>
<p>This is a fundamental operation in the theory of basic elements, see D. Eisenbud and E. G. Evans, <em>Basic elements: theorems from algebraic k-theory</em>, Bulletin of the AMS, <b>78</b>, No.4, 1972, 546-549.</p>
<div>Here is a basic example:</div>
<table class="examples"><tr><td><pre>i1 : kk=ZZ/32003
o1 = kk
o1 : QuotientRing</pre>
</td></tr>
<tr><td><pre>i2 : S=kk[a..d]
o2 = S
o2 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i3 : M=matrix{{a,0,0,0},{0,b,0,0},{0,0,c,0},{0,0,0,d}}
o3 = | a 0 0 0 |
| 0 b 0 0 |
| 0 0 c 0 |
| 0 0 0 d |
4 4
o3 : Matrix S <--- S</pre>
</td></tr>
<tr><td><pre>i4 : elementary(M,0,1)-- since k=0, this command simply eliminates the last row of M.
o4 = | a 0 0 0 |
| 0 b 0 0 |
| 0 0 c 0 |
3 4
o4 : Matrix S <--- S</pre>
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<div>Here is a more involved example. This is also how this function is used within the package.</div>
<table class="examples"><tr><td><pre>i5 : kk=ZZ/32003
o5 = kk
o5 : QuotientRing</pre>
</td></tr>
<tr><td><pre>i6 : S=kk[a..d]
o6 = S
o6 : PolynomialRing</pre>
</td></tr>
<tr><td><pre>i7 : I=ideal(a^2,b^3,c^4, d^5)
2 3 4 5
o7 = ideal (a , b , c , d )
o7 : Ideal of S</pre>
</td></tr>
<tr><td><pre>i8 : F=res I
1 4 6 4 1
o8 = S <-- S <-- S <-- S <-- S <-- 0
0 1 2 3 4 5
o8 : ChainComplex</pre>
</td></tr>
<tr><td><pre>i9 : M=image F.dd_3
o9 = image {5} | c4 d5 0 0 |
{6} | -b3 0 d5 0 |
{7} | a2 0 0 d5 |
{7} | 0 -b3 -c4 0 |
{8} | 0 a2 0 -c4 |
{9} | 0 0 a2 b3 |
6
o9 : S-module, submodule of S</pre>
</td></tr>
<tr><td><pre>i10 : f=matrix gens M
o10 = {5} | c4 d5 0 0 |
{6} | -b3 0 d5 0 |
{7} | a2 0 0 d5 |
{7} | 0 -b3 -c4 0 |
{8} | 0 a2 0 -c4 |
{9} | 0 0 a2 b3 |
6 4
o10 : Matrix S <--- S</pre>
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<tr><td><pre>i11 : fascending=transpose sort(transpose f, DegreeOrder=>Descending) -- this is f with rows sorted so that the degrees are ascending.
o11 = {5} | c4 d5 0 0 |
{6} | -b3 0 d5 0 |
{7} | 0 -b3 -c4 0 |
{7} | a2 0 0 d5 |
{8} | 0 a2 0 -c4 |
{9} | 0 0 a2 b3 |
6 4
o11 : Matrix S <--- S</pre>
</td></tr>
<tr><td><pre>i12 : g=elementary(fascending,1,1) --k=1, so add random multiples of the last row to the preceding row
o12 = {5} | c4 d5 0 0 |
{6} | -b3 0 d5 0 |
{7} | 0 -b3 -c4 0 |
{7} | a2 0 0 d5 |
{8} | 0 a2 10875a3 10875ab3-c4 |
5 4
o12 : Matrix S <--- S</pre>
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<tr><td><pre>i13 : g1=elementary(fascending,1,3)
o13 = {5} | c4 d5 0 0 |
{6} | -b3 0 d5 0 |
{7} | 0 -b3 -c4 0 |
{7} | a2 0 0 d5 |
{8} | 0 a2 15770a3-9599a2b-5559a2c 15770ab3-9599b4-5559b3c-c4 |
5 4
o13 : Matrix S <--- S</pre>
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<div>This method is called by <a href="_evans__Griffith.html" title="Reduces the rank of a syzygy">evansGriffith</a>.</div>
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<div class="waystouse"><h2>Ways to use <tt>elementary</tt> :</h2>
<ul><li>elementary(Matrix,ZZ,ZZ)</li>
</ul>
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