The computations performed in the routine
noetherNormalization use a random linear change of coordinates, hence one should expect the output to change each time the routine is executed.
i1 : R = QQ[x_1..x_4];
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i2 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);
o2 : Ideal of R
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i3 : (f,J,X) = noetherNormalization I
5 7 1 2 5
o3 = (map(R,R,{2x + -x + x , x , -x + -x + x , x }), ideal (3x + -x x +
1 3 2 4 1 4 1 4 2 3 2 1 3 1 2
------------------------------------------------------------------------
7 3 41 2 2 5 3 2 5 2 7 2
x x + 1, -x x + --x x + --x x + 2x x x + -x x x + -x x x +
1 4 2 1 2 12 1 2 12 1 2 1 2 3 3 1 2 3 4 1 2 4
------------------------------------------------------------------------
1 2
-x x x + x x x x + 1), {x , x })
4 1 2 4 1 2 3 4 4 3
o3 : Sequence
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The next example shows how when we use the lexicographical ordering, we can see the integrality of
R/ f I over the polynomial ring in
dim(R/I) variables:
i4 : R = QQ[x_1..x_5, MonomialOrder => Lex];
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i5 : I = ideal(x_2*x_1-x_5^3, x_5*x_1^3);
o5 : Ideal of R
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i6 : (f,J,X) = noetherNormalization I
5 1 1 1 8
o6 = (map(R,R,{-x + -x + x , x , -x + -x + x , -x + x + x , x }), ideal
8 1 8 2 5 1 2 1 2 2 4 9 1 2 3 2
------------------------------------------------------------------------
5 2 1 3 125 3 75 2 2 75 2 15 3
(-x + -x x + x x - x , ---x x + ---x x + --x x x + ---x x +
8 1 8 1 2 1 5 2 512 1 2 512 1 2 64 1 2 5 512 1 2
------------------------------------------------------------------------
15 2 15 2 1 4 3 3 3 2 2 3
--x x x + --x x x + ---x + --x x + -x x + x x ), {x , x , x })
32 1 2 5 8 1 2 5 512 2 64 2 5 8 2 5 2 5 5 4 3
o6 : Sequence
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i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 163840x_1x_2x_5^6-9600x_2^9x_5-5x_2^9+38400x_2^8x_5^2+40x_2^
{-9} | 40x_1x_2^2x_5^3-307200x_1x_2x_5^5+320x_1x_2x_5^4+18000x_2^9-
{-9} | 5x_1x_2^3+38400x_1x_2^2x_5^2+80x_1x_2^2x_5+188743680000x_1x_
{-3} | 5x_1^2+x_1x_2+8x_1x_5-8x_2^3
------------------------------------------------------------------------
8x_5-102400x_2^7x_5^3-320x_2^7x_5^2+2560x_2^6x_5^3-20480x_2
72000x_2^8x_5-25x_2^8+192000x_2^7x_5^2+400x_2^7x_5-4800x_2^
2x_5^5-98304000x_1x_2x_5^4+204800x_1x_2x_5^3+320x_1x_2x_5^2
------------------------------------------------------------------------
^5x_5^4+163840x_2^4x_5^5+32768x_2^2x_5^6+262144x_2x_5^7
6x_5^2+38400x_2^5x_5^3-307200x_2^4x_5^4+320x_2^4x_5^3+8x_2^3x_5^3-61440x
-11059200000x_2^9+44236800000x_2^8x_5+23040000x_2^8-117964800000x_2^7x_5
------------------------------------------------------------------------
_2^2x_5^5+128x_2^2x_5^4-491520x_2x_5^6+512x_2x_5^5
^2-307200000x_2^7x_5+64000x_2^7+2949120000x_2^6x_5^2-1536000x_2^6x_5-
------------------------------------------------------------------------
1600x_2^6-23592960000x_2^5x_5^3+12288000x_2^5x_5^2+12800x_2^5x_5+40x_2^5
------------------------------------------------------------------------
+188743680000x_2^4x_5^4-98304000x_2^4x_5^3+204800x_2^4x_5^2+320x_2^4x_5+
------------------------------------------------------------------------
x_2^4+7680x_2^3x_5^2+24x_2^3x_5+37748736000x_2^2x_5^5-19660800x_2^2x_5^4
------------------------------------------------------------------------
+102400x_2^2x_5^3+192x_2^2x_5^2+301989888000x_2x_5^6-157286400x_2x_5^5+
------------------------------------------------------------------------
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327680x_2x_5^4+512x_2x_5^3 |
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5 1
o7 : Matrix R <--- R
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If
noetherNormalization is unable to place the ideal into the desired position after a few tries, the following warning is given:
i8 : R = ZZ/2[a,b];
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i9 : I = ideal(a^2*b+a*b^2+1);
o9 : Ideal of R
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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
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Here is an example with the option
Verbose => true:
i11 : R = QQ[x_1..x_4];
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i12 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);
o12 : Ideal of R
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i13 : (f,J,X) = noetherNormalization(I,Verbose => true)
--trying random transformation: 1
--trying with basis element limit: 5
--trying with basis element limit: 20
2 5 9 2 5
o13 = (map(R,R,{-x + -x + x , x , 2x + 8x + x , x }), ideal (-x + -x x
7 1 9 2 4 1 1 2 3 2 7 1 9 1 2
-----------------------------------------------------------------------
4 3 214 2 2 40 3 2 2 5 2 2
+ x x + 1, -x x + ---x x + --x x + -x x x + -x x x + 2x x x +
1 4 7 1 2 63 1 2 9 1 2 7 1 2 3 9 1 2 3 1 2 4
-----------------------------------------------------------------------
2
8x x x + x x x x + 1), {x , x })
1 2 4 1 2 3 4 4 3
o13 : Sequence
|
The first number in the output above gives the number of linear transformations performed by the routine while attempting to place
I into the desired position. The second number tells which
BasisElementLimit was used when computing the (partial) Groebner basis. By default,
noetherNormalization tries to use a partial Groebner basis. It does this by sequentially computing a Groebner basis with the option
BasisElementLimit set to predetermined values. The default values come from the following list:
{5,20,40,60,80,infinity}. To set the values manually, use the option
LimitList:
i14 : R = QQ[x_1..x_4];
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i15 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);
o15 : Ideal of R
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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
1 5 1 1 3 2 5
o16 = (map(R,R,{-x + -x + x , x , --x + -x + x , x }), ideal (-x + -x x
2 1 2 2 4 1 10 1 2 2 3 2 2 1 2 1 2
-----------------------------------------------------------------------
1 3 1 2 2 5 3 1 2 5 2 1 2
+ x x + 1, --x x + -x x + -x x + -x x x + -x x x + --x x x +
1 4 20 1 2 2 1 2 4 1 2 2 1 2 3 2 1 2 3 10 1 2 4
-----------------------------------------------------------------------
1 2
-x x x + x x x x + 1), {x , x })
2 1 2 4 1 2 3 4 4 3
o16 : Sequence
|
To limit the randomness of the coefficients, use the option
RandomRange. Here is an example where the coefficients of the linear transformation are random integers from
-2 to
2:
i17 : R = QQ[x_1..x_4];
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i18 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);
o18 : Ideal of R
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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
2
o19 = (map(R,R,{3x + x , x , - 4x + x + x , x }), ideal (x + 3x x + x x
2 4 1 1 2 3 2 1 1 2 1 4
-----------------------------------------------------------------------
2 2 3 2 2 2
+ 1, - 12x x + 3x x + 3x x x - 4x x x + x x x + x x x x + 1),
1 2 1 2 1 2 3 1 2 4 1 2 4 1 2 3 4
-----------------------------------------------------------------------
{x , x })
4 3
o19 : Sequence
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This symbol is provided by the package NoetherNormalization.