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
2 7 8 1 11 2 7
o3 = (map(R,R,{-x + -x + x , x , -x + --x + x , x }), ideal (--x + -x x
9 1 3 2 4 1 3 1 10 2 3 2 9 1 3 1 2
------------------------------------------------------------------------
16 3 281 2 2 7 3 2 2 7 2 8 2
+ x x + 1, --x x + ---x x + --x x + -x x x + -x x x + -x x x +
1 4 27 1 2 45 1 2 30 1 2 9 1 2 3 3 1 2 3 3 1 2 4
------------------------------------------------------------------------
1 2
--x x x + x x x x + 1), {x , x })
10 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
8 10 1 1 8
o6 = (map(R,R,{-x + --x + x , x , -x + 6x + x , -x + -x + x , x }),
9 1 3 2 5 1 2 1 2 4 5 1 3 2 3 2
------------------------------------------------------------------------
8 2 10 3 512 3 640 2 2 64 2 800 3
ideal (-x + --x x + x x - x , ---x x + ---x x + --x x x + ---x x
9 1 3 1 2 1 5 2 729 1 2 81 1 2 27 1 2 5 27 1 2
------------------------------------------------------------------------
160 2 8 2 1000 4 100 3 2 2 3
+ ---x x x + -x x x + ----x + ---x x + 10x x + x x ), {x , x , x })
9 1 2 5 3 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} | 1944x_1x_2x_5^6-115200x_2^9x_5-800000x_2^9+17280x_2^8x_5^2+
{-9} | 90000x_1x_2^2x_5^3-1944x_1x_2x_5^5+27000x_1x_2x_5^4+115200x
{-9} | 1562500000x_1x_2^3+33750000x_1x_2^2x_5^2+937500000x_1x_2^2x
{-3} | 8x_1^2+30x_1x_2+9x_1x_5-9x_2^3
------------------------------------------------------------------------
240000x_2^8x_5-1728x_2^7x_5^3-72000x_2^7x_5^2+21600x_2^6x_5^3-6480x_2^
_2^9-17280x_2^8x_5-80000x_2^8+1728x_2^7x_5^2+48000x_2^7x_5-21600x_2^6x
_5+34992x_1x_2x_5^5-243000x_1x_2x_5^4+6750000x_1x_2x_5^3+140625000x_1x
------------------------------------------------------------------------
5x_5^4+1944x_2^4x_5^5+7290x_2^2x_5^6+2187x_2x_5^7
_5^2+6480x_2^5x_5^3-1944x_2^4x_5^4+27000x_2^4x_5^3+337500x_2^3x_5
_2x_5^2-2073600x_2^9+311040x_2^8x_5+2160000x_2^8-31104x_2^7x_5^2-
------------------------------------------------------------------------
^3-7290x_2^2x_5^5+202500x_2^2x_5^4-2187x_2x_5^6+30375x_2x_5^5
1080000x_2^7x_5+3000000x_2^7+388800x_2^6x_5^2-2700000x_2^6x_5-37500000x_
------------------------------------------------------------------------
2^6-116640x_2^5x_5^3+810000x_2^5x_5^2+11250000x_2^5x_5+468750000x_2^5+
------------------------------------------------------------------------
34992x_2^4x_5^4-243000x_2^4x_5^3+6750000x_2^4x_5^2+140625000x_2^4x_5+
------------------------------------------------------------------------
5859375000x_2^4+126562500x_2^3x_5^2+5273437500x_2^3x_5+131220x_2^2x_5^5-
------------------------------------------------------------------------
911250x_2^2x_5^4+63281250x_2^2x_5^3+1582031250x_2^2x_5^2+39366x_2x_5^6-
------------------------------------------------------------------------
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273375x_2x_5^5+7593750x_2x_5^4+158203125x_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
7 3 7 2 7
o13 = (map(R,R,{x + -x + x , x , -x + --x + x , x }), ideal (2x + -x x
1 3 2 4 1 8 1 10 2 3 2 1 3 1 2
-----------------------------------------------------------------------
3 3 63 2 2 49 3 2 7 2 3 2
+ x x + 1, -x x + --x x + --x x + x x x + -x x x + -x x x +
1 4 8 1 2 40 1 2 30 1 2 1 2 3 3 1 2 3 8 1 2 4
-----------------------------------------------------------------------
7 2
--x x x + x x x x + 1), {x , x })
10 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
5 5 3 7 2 5
o16 = (map(R,R,{-x + -x + x , x , -x + x + x , x }), ideal (-x + -x x +
2 1 2 2 4 1 4 1 2 3 2 2 1 2 1 2
-----------------------------------------------------------------------
15 3 35 2 2 5 3 5 2 5 2 3 2
x x + 1, --x x + --x x + -x x + -x x x + -x x x + -x x x +
1 4 8 1 2 8 1 2 2 1 2 2 1 2 3 2 1 2 3 4 1 2 4
-----------------------------------------------------------------------
2
x x x + x x x x + 1), {x , x })
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
--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
--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: 4
--trying with basis element limit: 5
--trying with basis element limit: 20
2
o19 = (map(R,R,{- 8x - 5x + x , x , 8x + x + x , x }), ideal (- 7x -
1 2 4 1 1 2 3 2 1
-----------------------------------------------------------------------
3 2 2 3 2 2
5x x + x x + 1, - 64x x - 48x x - 5x x - 8x x x - 5x x x +
1 2 1 4 1 2 1 2 1 2 1 2 3 1 2 3
-----------------------------------------------------------------------
2 2
8x x x + x 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.