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<head><title>schreyerOrder(Matrix) -- create a matrix with the same entries whose source free module has a Schreyer monomial order</title>
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<div><h1>schreyerOrder(Matrix) -- create a matrix with the same entries whose source free module has a Schreyer monomial order</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>schreyerOrder m</tt></div>
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<li><span>Function: <a href="_schreyer__Order.html" title="create or obtain free modules with Schryer monomial orders">schreyerOrder</a></span></li>
<li><div class="single">Inputs:<ul><li><span><tt>m</tt>, <span>a <a href="___Matrix.html">matrix</a></span>, G <-- F between free modules</span></li>
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<li><div class="single">Outputs:<ul><li><span><span>a <a href="___Matrix.html">matrix</a></span>, the same matrix as <tt>m</tt>, except that its source free module is endowed with a Schreyer, or induced, monomial order</span></li>
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<div class="single"><h2>Description</h2>
<div>Given a matrix <i>m : F --> G</i>, the Schreyer order on the monomials of F is given by: If <i>a e<sub>i</sub></i> and <i>b e<sub>j</sub></i> are monomials of <i>F</i>, i.e. <i>a</i> and <i>b</i> are monomials in the ring, and <i>e<sub>i</sub></i> and <i>e<sub>j</sub></i> are unit column vectors of <i>F</i>, then <i>a e<sub>i</sub> > b e<sub>j</sub></i> if and only if either <i>leadterm(m)(a e<sub>i</sub>) > leadterm(m)(b e<sub>j</sub>)</i> or they are scalar multiples of the same monomial in <i>G</i>, and <i>i > j</i>.<p/>
If the base ring is a quotient ring, we think of <tt>leadterm(m)</tt> as a matrix over the ambient polynomial ring for the purpose of this definition.<p/>
In the example below, the source of <tt>f</tt> is endowed with a Schreyer order.<table class="examples"><tr><td><pre>i1 : R = ZZ/101[a..d];</pre>
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<tr><td><pre>i2 : m = matrix{{a,b,c,d}};
1 4
o2 : Matrix R <--- R</pre>
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<tr><td><pre>i3 : f = schreyerOrder m
o3 = | a b c d |
1 4
o3 : Matrix R <--- R</pre>
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<tr><td><pre>i4 : g = syz f
o4 = {1} | -b 0 -c 0 0 -d |
{1} | a -c 0 0 -d 0 |
{1} | 0 b a -d 0 0 |
{1} | 0 0 0 c b a |
4 6
o4 : Matrix R <--- R</pre>
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<tr><td><pre>i5 : leadTerm g
o5 = {1} | 0 0 0 0 0 0 |
{1} | a 0 0 0 0 0 |
{1} | 0 b a 0 0 0 |
{1} | 0 0 0 c b a |
4 6
o5 : Matrix R <--- R</pre>
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<tr><td><pre>i6 : hf = map(source f, 1, {{d},{c},{b},{a}})
o6 = {1} | d |
{1} | c |
{1} | b |
{1} | a |
4 1
o6 : Matrix R <--- R</pre>
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<tr><td><pre>i7 : hm = map(source m, 1, {{d},{c},{b},{a}})
o7 = {1} | d |
{1} | c |
{1} | b |
{1} | a |
4 1
o7 : Matrix R <--- R</pre>
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<tr><td><pre>i8 : leadTerm hf
o8 = {1} | 0 |
{1} | 0 |
{1} | b |
{1} | 0 |
4 1
o8 : Matrix R <--- R</pre>
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<tr><td><pre>i9 : leadTerm hm
o9 = {1} | 0 |
{1} | 0 |
{1} | 0 |
{1} | a |
4 1
o9 : Matrix R <--- R</pre>
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Use <a href="_schreyer__Order_lp__Module_rp.html" title="obtain Schreyer order information">schreyerOrder(Module)</a> to see if a free module is endowed with a Schreyer order.<table class="examples"><tr><td><pre>i10 : schreyerOrder source m
o10 = 0
4
o10 : Matrix 0 <--- R</pre>
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<tr><td><pre>i11 : schreyerOrder source f
o11 = | a 0 0 0 |
| 0 b 0 0 |
| 0 0 c 0 |
| 0 0 0 d |
4 4
o11 : Matrix R <--- R</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="___Schreyer_sporders.html" title="induced monomial order on a free module">Schreyer orders</a> -- induced monomial order on a free module</span></li>
<li><span><a href="_monomial_sporders_spfor_spfree_spmodules.html" title="">monomial orders for free modules</a></span></li>
<li><span><a href="_lead__Term.html" title="get the greatest term">leadTerm</a> -- get the greatest term</span></li>
<li><span><a href="_schreyer__Order_lp__Module_rp.html" title="obtain Schreyer order information">schreyerOrder(Module)</a> -- obtain Schreyer order information</span></li>
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