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
4 1 4 1 11 2 1
o3 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (--x + -x x
7 1 2 2 4 1 7 1 3 2 3 2 7 1 2 1 2
------------------------------------------------------------------------
16 3 10 2 2 1 3 4 2 1 2 4 2
+ x x + 1, --x x + --x x + -x x + -x x x + -x x x + -x x x +
1 4 49 1 2 21 1 2 6 1 2 7 1 2 3 2 1 2 3 7 1 2 4
------------------------------------------------------------------------
1 2
-x x x + x x x x + 1), {x , x })
3 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 7
o6 = (map(R,R,{5x + -x + x , x , -x + 10x + x , x + x + x , x }), ideal
1 3 2 5 1 3 1 2 4 1 2 3 2
------------------------------------------------------------------------
2 5 3 3 2 2 2 125 3
(5x + -x x + x x - x , 125x x + 125x x + 75x x x + ---x x +
1 3 1 2 1 5 2 1 2 1 2 1 2 5 3 1 2
------------------------------------------------------------------------
2 2 125 4 25 3 2 2 3
50x x x + 15x x x + ---x + --x x + 5x x + x x ), {x , x , x })
1 2 5 1 2 5 27 2 3 2 5 2 5 2 5 5 4 3
o6 : Sequence
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i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 1215x_1x_2x_5^6-101250x_2^9x_5-15625x_2^9+30375x_2^8x_5^2
{-9} | 1875x_1x_2^2x_5^3-3645x_1x_2x_5^5+1125x_1x_2x_5^4+303750x
{-9} | 5859375x_1x_2^3+11390625x_1x_2^2x_5^2+7031250x_1x_2^2x_5+
{-3} | 15x_1^2+5x_1x_2+3x_1x_5-3x_2^3
------------------------------------------------------------------------
+9375x_2^8x_5-6075x_2^7x_5^3-5625x_2^7x_5^2+3375x_2^6x_5^3-2025x_2^5x_5^
_2^9-91125x_2^8x_5-9375x_2^8+18225x_2^7x_5^2+11250x_2^7x_5-10125x_2^6x_5
47829690x_1x_2x_5^5-7381125x_1x_2x_5^4+4556250x_1x_2x_5^3+2109375x_1x_2x
------------------------------------------------------------------------
4+1215x_2^4x_5^5+405x_2^2x_5^6+243x_2x_5^7
^2+6075x_2^5x_5^3-3645x_2^4x_5^4+1125x_2^4x_5^3+625x_2^3x_5^3-1215x_2^2x
_5^2-3985807500x_2^9+1195742250x_2^8x_5+184528125x_2^8-239148450x_2^7x_5
------------------------------------------------------------------------
_5^5+750x_2^2x_5^4-729x_2x_5^6+225x_2x_5^5
^2-184528125x_2^7x_5+11390625x_2^7+132860250x_2^6x_5^2-20503125x_2^6x_5-
------------------------------------------------------------------------
6328125x_2^6-79716150x_2^5x_5^3+12301875x_2^5x_5^2+3796875x_2^5x_5+
------------------------------------------------------------------------
3515625x_2^5+47829690x_2^4x_5^4-7381125x_2^4x_5^3+4556250x_2^4x_5^2+
------------------------------------------------------------------------
2109375x_2^4x_5+1953125x_2^4+3796875x_2^3x_5^2+3515625x_2^3x_5+15943230x
------------------------------------------------------------------------
_2^2x_5^5-2460375x_2^2x_5^4+3796875x_2^2x_5^3+2109375x_2^2x_5^2+9565938x
------------------------------------------------------------------------
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_2x_5^6-1476225x_2x_5^5+911250x_2x_5^4+421875x_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,{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
4 2
o13 = (map(R,R,{2x + 10x + x , x , x + -x + x , x }), ideal (3x + 10x x
1 2 4 1 1 9 2 3 2 1 1 2
-----------------------------------------------------------------------
3 98 2 2 40 3 2 2 2
+ x x + 1, 2x x + --x x + --x x + 2x x x + 10x x x + x x x +
1 4 1 2 9 1 2 9 1 2 1 2 3 1 2 3 1 2 4
-----------------------------------------------------------------------
4 2
-x x x + x x x x + 1), {x , x })
9 1 2 4 1 2 3 4 4 3
o13 : Sequence
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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
3 2 3
o16 = (map(R,R,{x + -x + x , x , 4x + x + x , x }), ideal (2x + -x x +
1 5 2 4 1 1 2 3 2 1 5 1 2
-----------------------------------------------------------------------
3 17 2 2 3 3 2 3 2 2 2
x x + 1, 4x x + --x x + -x x + x x x + -x x x + 4x x x + x x x
1 4 1 2 5 1 2 5 1 2 1 2 3 5 1 2 3 1 2 4 1 2 4
-----------------------------------------------------------------------
+ x x x x + 1), {x , x })
1 2 3 4 4 3
o16 : Sequence
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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 + 2x + x , x , 6x + 2x + x , x }), ideal (- 2x +
1 2 4 1 1 2 3 2 1
-----------------------------------------------------------------------
3 2 2 3 2 2
2x x + x x + 1, - 18x x + 6x x + 4x x - 3x x x + 2x x x +
1 2 1 4 1 2 1 2 1 2 1 2 3 1 2 3
-----------------------------------------------------------------------
2 2
6x x x + 2x x x + x x x x + 1), {x , x })
1 2 4 1 2 4 1 2 3 4 4 3
o19 : Sequence
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This symbol is provided by the package NoetherNormalization.