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<head><title>noetherNormalization -- data for Noether normalization</title>
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<div><h1>noetherNormalization -- data for Noether normalization</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>(f,J,X) = noetherNormalization C</tt></div>
</dd></dl>
</div>
</li>
<li><div class="single">Inputs:<ul><li><span><tt>C</tt>, which is either <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span> <tt>I</tt>, or <span>a <a href="../../Macaulay2Doc/html/___Quotient__Ring.html">quotient ring</a></span> <tt>R/I</tt> where <tt>R</tt> is <span>a <a href="../../Macaulay2Doc/html/___Polynomial__Ring.html">polynomial ring</a></span></span></li>
<li><span><a href="../../Macaulay2Doc/html/___Basis__Element__Limit.html" title="name for an optional argument">BasisElementLimit</a></span></li>
<li><span>will take</span></li>
</ul>
</div>
</li>
<li><div class="single">Outputs:<ul><li><span><tt>f</tt>, <span>a <a href="../../Macaulay2Doc/html/___Ring__Map.html">ring map</a></span>, an automorphism of <tt>R</tt></span></li>
<li><span><tt>J</tt>, <span>an <a href="../../Macaulay2Doc/html/___Ideal.html">ideal</a></span>, the image of <tt>I</tt> under <tt>f</tt></span></li>
<li><span><tt>X</tt>, <span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, a list of variables which are algebraically independent in <tt>R/J</tt></span></li>
</ul>
</div>
</li>
<li><div class="single"><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>:<ul><li><span><tt>LimitList => </tt><span><span>a <a href="../../Macaulay2Doc/html/___List.html">list</a></span>, <span>default value {5, 20, 40, 60, 80, infinity}</span>, gives the value which </span></span></li>
<li><span><tt>RandomRange => </tt><span><span>an <a href="../../Macaulay2Doc/html/___Z__Z.html">integer</a></span>, <span>default value 0</span>, if not 0, gives a integer bound for the random coefficients. If 0, then chooses random elements from the coefficient field.</span></span></li>
<li><span><a href="_noether__Normalization.html">Verbose => ...</a>, -- data for Noether normalization</span></li>
</ul>
</div>
</li>
</ul>
</div>
<div class="single"><h2>Description</h2>
<div>The computations performed in the routine <tt>noetherNormalization</tt> use a random linear change of coordinates, hence one should expect the output to change each time the routine is executed.<table class="examples"><tr><td><pre>i1 : R = QQ[x_1..x_4];</pre>
</td></tr>
<tr><td><pre>i2 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);
o2 : Ideal of R</pre>
</td></tr>
<tr><td><pre>i3 : (f,J,X) = noetherNormalization I
1 2 1
o3 = (map(R,R,{x + --x + x , x , 8x + 3x + x , x }), ideal (2x + --x x
1 10 2 4 1 1 2 3 2 1 10 1 2
------------------------------------------------------------------------
3 19 2 2 3 3 2 1 2 2
+ x x + 1, 8x x + --x x + --x x + x x x + --x x x + 8x x x +
1 4 1 2 5 1 2 10 1 2 1 2 3 10 1 2 3 1 2 4
------------------------------------------------------------------------
2
3x x x + x x x x + 1), {x , x })
1 2 4 1 2 3 4 4 3
o3 : Sequence</pre>
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The next example shows how when we use the lexicographical ordering, we can see the integrality of <tt>R/ f I</tt> over the polynomial ring in <tt>dim(R/I)</tt> variables:<table class="examples"><tr><td><pre>i4 : R = QQ[x_1..x_5, MonomialOrder => Lex];</pre>
</td></tr>
<tr><td><pre>i5 : I = ideal(x_2*x_1-x_5^3, x_5*x_1^3);
o5 : Ideal of R</pre>
</td></tr>
<tr><td><pre>i6 : (f,J,X) = noetherNormalization I
9 7
o6 = (map(R,R,{-x + 9x + x , x , x + x + x , -x + 9x + x , x }), ideal
5 1 2 5 1 1 2 4 8 1 2 3 2
------------------------------------------------------------------------
9 2 3 729 3 2187 2 2 243 2 2187 3
(-x + 9x x + x x - x , ---x x + ----x x + ---x x x + ----x x +
5 1 1 2 1 5 2 125 1 2 25 1 2 25 1 2 5 5 1 2
------------------------------------------------------------------------
486 2 27 2 4 3 2 2 3
---x x x + --x x x + 729x + 243x x + 27x x + x x ), {x , x , x })
5 1 2 5 5 1 2 5 2 2 5 2 5 2 5 5 4 3
o6 : Sequence</pre>
</td></tr>
<tr><td><pre>i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 45x_1x_2x_5^6-39366x_2^9x_5-2657205x_2^9+2187x_2^8x_5^2+
{-9} | 54675x_1x_2^2x_5^3-45x_1x_2x_5^5+6075x_1x_2x_5^4+39366x_
{-9} | 26904200625x_1x_2^3+22143375x_1x_2^2x_5^2+5978711250x_1x
{-3} | 9x_1^2+45x_1x_2+5x_1x_5-5x_2^3
------------------------------------------------------------------------
295245x_2^8x_5-81x_2^7x_5^3-32805x_2^7x_5^2+3645x_2^6x_5^3-405x_2^5x_5
2^9-2187x_2^8x_5-98415x_2^8+81x_2^7x_5^2+21870x_2^7x_5-3645x_2^6x_5^2+
_2^2x_5+90x_1x_2x_5^5-6075x_1x_2x_5^4+1640250x_1x_2x_5^3+332150625x_1x
------------------------------------------------------------------------
^4+45x_2^4x_5^5+225x_2^2x_5^6+25x_2x_5^7
405x_2^5x_5^3-45x_2^4x_5^4+6075x_2^4x_5^3+273375x_2^3x_5^3-225x_2^2x_5^5
_2x_5^2-78732x_2^9+4374x_2^8x_5+295245x_2^8-162x_2^7x_5^2-54675x_2^7x_5+
------------------------------------------------------------------------
+60750x_2^2x_5^4-25x_2x_5^6+3375x_2x_5^5
1476225x_2^7+7290x_2^6x_5^2-492075x_2^6x_5-66430125x_2^6-810x_2^5x_5^3+
------------------------------------------------------------------------
54675x_2^5x_5^2+7381125x_2^5x_5+2989355625x_2^5+90x_2^4x_5^4-6075x_2^4x_
------------------------------------------------------------------------
5^3+1640250x_2^4x_5^2+332150625x_2^4x_5+134521003125x_2^4+110716875x_2^
------------------------------------------------------------------------
3x_5^2+44840334375x_2^3x_5+450x_2^2x_5^5-30375x_2^2x_5^4+20503125x_2^2x_
------------------------------------------------------------------------
5^3+4982259375x_2^2x_5^2+50x_2x_5^6-3375x_2x_5^5+911250x_2x_5^4+
------------------------------------------------------------------------
|
|
|
184528125x_2x_5^3 |
|
5 1
o7 : Matrix R <--- R</pre>
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</table>
If <tt>noetherNormalization</tt> is unable to place the ideal into the desired position after a few tries, the following warning is given:<table class="examples"><tr><td><pre>i8 : R = ZZ/2[a,b];</pre>
</td></tr>
<tr><td><pre>i9 : I = ideal(a^2*b+a*b^2+1);
o9 : Ideal of R</pre>
</td></tr>
<tr><td><pre>i10 : (f,J,X) = noetherNormalization I
--warning: no good linear transformation found by noetherNormalization
2 2
o10 = (map(R,R,{a + b, a}), ideal(a b + a*b + 1), {b})
o10 : Sequence</pre>
</td></tr>
</table>
Here is an example with the option <tt>Verbose => true</tt>:<table class="examples"><tr><td><pre>i11 : R = QQ[x_1..x_4];</pre>
</td></tr>
<tr><td><pre>i12 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);
o12 : Ideal of R</pre>
</td></tr>
<tr><td><pre>i13 : (f,J,X) = noetherNormalization(I,Verbose => true)
--trying random transformation: 1
--trying with basis element limit: 5
--trying with basis element limit: 20
1 3 5 6 11 2
o13 = (map(R,R,{--x + -x + x , x , -x + -x + x , x }), ideal (--x +
10 1 2 2 4 1 3 1 7 2 3 2 10 1
-----------------------------------------------------------------------
3 1 3 181 2 2 9 3 1 2 3 2
-x x + x x + 1, -x x + ---x x + -x x + --x x x + -x x x +
2 1 2 1 4 6 1 2 70 1 2 7 1 2 10 1 2 3 2 1 2 3
-----------------------------------------------------------------------
5 2 6 2
-x x x + -x x x + x x x x + 1), {x , x })
3 1 2 4 7 1 2 4 1 2 3 4 4 3
o13 : Sequence</pre>
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</table>
The first number in the output above gives the number of linear transformations performed by the routine while attempting to place <tt>I</tt> into the desired position. The second number tells which <a href="../../Macaulay2Doc/html/___Basis__Element__Limit.html" title="name for an optional argument">BasisElementLimit</a> was used when computing the (partial) Groebner basis. By default, <tt>noetherNormalization</tt> tries to use a partial Groebner basis. It does this by sequentially computing a Groebner basis with the option <a href="../../Macaulay2Doc/html/___Basis__Element__Limit.html" title="name for an optional argument">BasisElementLimit</a> set to predetermined values. The default values come from the following list:<tt>{5,20,40,60,80,infinity}</tt>. To set the values manually, use the option <tt>LimitList</tt>:<table class="examples"><tr><td><pre>i14 : R = QQ[x_1..x_4]; </pre>
</td></tr>
<tr><td><pre>i15 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);
o15 : Ideal of R</pre>
</td></tr>
<tr><td><pre>i16 : (f,J,X) = noetherNormalization(I,Verbose => true,LimitList => {5,10})
--trying random transformation: 1
--trying with basis element limit: 5
--trying with basis element limit: 10
7 10 4 16 2
o16 = (map(R,R,{-x + --x + x , x , 4x + -x + x , x }), ideal (--x +
9 1 3 2 4 1 1 3 2 3 2 9 1
-----------------------------------------------------------------------
10 28 3 388 2 2 40 3 7 2 10 2
--x x + x x + 1, --x x + ---x x + --x x + -x x x + --x x x +
3 1 2 1 4 9 1 2 27 1 2 9 1 2 9 1 2 3 3 1 2 3
-----------------------------------------------------------------------
2 4 2
4x x x + -x x x + x x x x + 1), {x , x })
1 2 4 3 1 2 4 1 2 3 4 4 3
o16 : Sequence</pre>
</td></tr>
</table>
To limit the randomness of the coefficients, use the option <tt>RandomRange</tt>. Here is an example where the coefficients of the linear transformation are random integers from <tt>-2</tt> to <tt>2</tt>:<table class="examples"><tr><td><pre>i17 : R = QQ[x_1..x_4];</pre>
</td></tr>
<tr><td><pre>i18 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);
o18 : Ideal of R</pre>
</td></tr>
<tr><td><pre>i19 : (f,J,X) = noetherNormalization(I,Verbose => true,RandomRange => 2)
--trying random transformation: 1
--trying with basis element limit: 5
--trying with basis element limit: 20
--trying with basis element limit: 40
--trying with basis element limit: 60
--trying with basis element limit: 80
--trying with basis element limit: infinity
--trying random transformation: 2
--trying with basis element limit: 5
--trying with basis element limit: 20
--trying with basis element limit: 40
--trying with basis element limit: 60
--trying with basis element limit: 80
--trying with basis element limit: infinity
--trying random transformation: 3
--trying with basis element limit: 5
--trying with basis element limit: 20
2
o19 = (map(R,R,{6x + 3x + x , x , 3x + x + x , x }), ideal (7x + 3x x +
1 2 4 1 1 2 3 2 1 1 2
-----------------------------------------------------------------------
3 2 2 3 2 2 2
x x + 1, 18x x + 15x x + 3x x + 6x x x + 3x x x + 3x x x +
1 4 1 2 1 2 1 2 1 2 3 1 2 3 1 2 4
-----------------------------------------------------------------------
2
x x x + x x x x + 1), {x , x })
1 2 4 1 2 3 4 4 3
o19 : Sequence</pre>
</td></tr>
</table>
<p>This symbol is provided by the package <a href="index.html" title="routines related to Noether normalization">NoetherNormalization</a>.</p>
</div>
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<div class="waystouse"><h2>Ways to use <tt>noetherNormalization</tt> :</h2>
<ul><li>noetherNormalization(Ideal)</li>
<li>noetherNormalization(PolynomialRing)</li>
<li>noetherNormalization(QuotientRing)</li>
</ul>
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