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
1 3 7 4 2 3
o3 = (map(R,R,{-x + -x + x , x , -x + 9x + x , x }), ideal (-x + -x x +
3 1 5 2 4 1 4 1 2 3 2 3 1 5 1 2
------------------------------------------------------------------------
7 3 81 2 2 27 3 1 2 3 2 7 2
x x + 1, --x x + --x x + --x x + -x x x + -x x x + -x x x +
1 4 12 1 2 20 1 2 5 1 2 3 1 2 3 5 1 2 3 4 1 2 4
------------------------------------------------------------------------
2
9x x x + x x x x + 1), {x , x })
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
4 9 1 3
o6 = (map(R,R,{-x + -x + x , x , x + 10x + x , -x + -x + x , x }),
5 1 5 2 5 1 1 2 4 5 1 8 2 3 2
------------------------------------------------------------------------
4 2 9 3 64 3 432 2 2 48 2 972 3
ideal (-x + -x x + x x - x , ---x x + ---x x + --x x x + ---x x +
5 1 5 1 2 1 5 2 125 1 2 125 1 2 25 1 2 5 125 1 2
------------------------------------------------------------------------
216 2 12 2 729 4 243 3 27 2 2 3
---x x x + --x x x + ---x + ---x x + --x x + x x ), {x , x , x })
25 1 2 5 5 1 2 5 125 2 25 2 5 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} | 12500x_1x_2x_5^6-194400x_2^9x_5-236196x_2^9+54000x_2^8x_5^2+
{-9} | 43740x_1x_2^2x_5^3-10000x_1x_2x_5^5+24300x_1x_2x_5^4+155520x
{-9} | 13947137604x_1x_2^3+3188646000x_1x_2^2x_5^2+15496819560x_1x_
{-3} | 4x_1^2+9x_1x_2+5x_1x_5-5x_2^3
------------------------------------------------------------------------
131220x_2^8x_5-10000x_2^7x_5^3-72900x_2^7x_5^2+40500x_2^6x_5^3-22500x_2
_2^9-43200x_2^8x_5-34992x_2^8+8000x_2^7x_5^2+38880x_2^7x_5-32400x_2^6x_
2^2x_5+200000000x_1x_2x_5^5-243000000x_1x_2x_5^4+1180980000x_1x_2x_5^3+
------------------------------------------------------------------------
^5x_5^4+12500x_2^4x_5^5+28125x_2^2x_5^6+15625x_2x_5^7
5^2+18000x_2^5x_5^3-10000x_2^4x_5^4+24300x_2^4x_5^3+98415x_2^3x_5^3-
4304672100x_1x_2x_5^2-3110400000x_2^9+864000000x_2^8x_5+1049760000x_
------------------------------------------------------------------------
22500x_2^2x_5^5+109350x_2^2x_5^4-12500x_2x_5^6+30375x_2x_5^5
2^8-160000000x_2^7x_5^2-972000000x_2^7x_5+472392000x_2^7+648000000x_2^6x
------------------------------------------------------------------------
_5^2-787320000x_2^6x_5-1913187600x_2^6-360000000x_2^5x_5^3+437400000x_2^
------------------------------------------------------------------------
5x_5^2+1062882000x_2^5x_5+7748409780x_2^5+200000000x_2^4x_5^4-243000000x
------------------------------------------------------------------------
_2^4x_5^3+1180980000x_2^4x_5^2+4304672100x_2^4x_5+31381059609x_2^4+
------------------------------------------------------------------------
7174453500x_2^3x_5^2+52301766015x_2^3x_5+450000000x_2^2x_5^5-546750000x_
------------------------------------------------------------------------
2^2x_5^4+6643012500x_2^2x_5^3+29056536675x_2^2x_5^2+250000000x_2x_5^6-
------------------------------------------------------------------------
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303750000x_2x_5^5+1476225000x_2x_5^4+5380840125x_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
2
o13 = (map(R,R,{3x + 4x + x , x , x + 2x + x , x }), ideal (4x + 4x x +
1 2 4 1 1 2 3 2 1 1 2
-----------------------------------------------------------------------
3 2 2 3 2 2 2 2
x x + 1, 3x x + 10x x + 8x x + 3x x x + 4x x x + x x x + 2x x x
1 4 1 2 1 2 1 2 1 2 3 1 2 3 1 2 4 1 2 4
-----------------------------------------------------------------------
+ x x x x + 1), {x , x })
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 4 9 7 2 4
o16 = (map(R,R,{-x + -x + x , x , -x + 2x + x , x }), ideal (-x + -x x
2 1 3 2 4 1 5 1 2 3 2 2 1 3 1 2
-----------------------------------------------------------------------
9 3 37 2 2 8 3 5 2 4 2 9 2
+ x x + 1, -x x + --x x + -x x + -x x x + -x x x + -x x x +
1 4 2 1 2 5 1 2 3 1 2 2 1 2 3 3 1 2 3 5 1 2 4
-----------------------------------------------------------------------
2
2x 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
2
o19 = (map(R,R,{x + x , x , x + 2x + x , x }), ideal (x + x x + x x +
2 4 1 1 2 3 2 1 1 2 1 4
-----------------------------------------------------------------------
2 2 3 2 2 2
1, x x + 2x x + x x x + x x x + 2x x x + x x x x + 1), {x , x })
1 2 1 2 1 2 3 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.