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 3 2
o3 = (map(R,R,{-x + 7x + x , x , -x + 2x + x , x }), ideal (-x + 7x x +
2 1 2 4 1 5 1 2 3 2 2 1 1 2
------------------------------------------------------------------------
3 3 26 2 2 3 1 2 2 3 2
x x + 1, --x x + --x x + 14x x + -x x x + 7x x x + -x x x +
1 4 10 1 2 5 1 2 1 2 2 1 2 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
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
3 3 1 10 9
o6 = (map(R,R,{-x + -x + x , x , -x + 5x + x , --x + -x + x , x }),
8 1 5 2 5 1 7 1 2 4 3 1 4 2 3 2
------------------------------------------------------------------------
3 2 3 3 27 3 81 2 2 27 2 81 3
ideal (-x + -x x + x x - x , ---x x + ---x x + --x x x + ---x x +
8 1 5 1 2 1 5 2 512 1 2 320 1 2 64 1 2 5 200 1 2
------------------------------------------------------------------------
27 2 9 2 27 4 27 3 9 2 2 3
--x x x + -x x x + ---x + --x x + -x x + x x ), {x , x , x })
20 1 2 5 8 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} | 75000x_1x_2x_5^6-60750x_2^9x_5-5832x_2^9+50625x_2^8x
{-9} | 8640x_1x_2^2x_5^3-75000x_1x_2x_5^5+14400x_1x_2x_5^4+
{-9} | 22394880x_1x_2^3+194400000x_1x_2^2x_5^2+74649600x_1x
{-3} | 15x_1^2+24x_1x_2+40x_1x_5-40x_2^3
------------------------------------------------------------------------
_5^2+9720x_2^8x_5-28125x_2^7x_5^3-16200x_2^7x_5^2+27000x_2^6x_5^3-45000x
60750x_2^9-50625x_2^8x_5-3240x_2^8+28125x_2^7x_5^2+10800x_2^7x_5-27000x_
_2^2x_5+5859375000x_1x_2x_5^5-562500000x_1x_2x_5^4+216000000x_1x_2x_5^3+
------------------------------------------------------------------------
_2^5x_5^4+75000x_2^4x_5^5+120000x_2^2x_5^6+200000x_2x_5^7
2^6x_5^2+45000x_2^5x_5^3-75000x_2^4x_5^4+14400x_2^4x_5^3+13824x_2^3x_5
62208000x_1x_2x_5^2-4746093750x_2^9+3955078125x_2^8x_5+379687500x_2^8-
------------------------------------------------------------------------
^3-120000x_2^2x_5^5+46080x_2^2x_5^4-200000x_2x_5^6+38400x_2x_5^5
2197265625x_2^7x_5^2-1054687500x_2^7x_5+40500000x_2^7+2109375000x_2^6x_5
------------------------------------------------------------------------
^2-202500000x_2^6x_5-38880000x_2^6-3515625000x_2^5x_5^3+337500000x_2^5x_
------------------------------------------------------------------------
5^2+64800000x_2^5x_5+37324800x_2^5+5859375000x_2^4x_5^4-562500000x_2^4x_
------------------------------------------------------------------------
5^3+216000000x_2^4x_5^2+62208000x_2^4x_5+35831808x_2^4+311040000x_2^3x_5
------------------------------------------------------------------------
^2+179159040x_2^3x_5+9375000000x_2^2x_5^5-900000000x_2^2x_5^4+864000000x
------------------------------------------------------------------------
_2^2x_5^3+298598400x_2^2x_5^2+15625000000x_2x_5^6-1500000000x_2x_5^5+
------------------------------------------------------------------------
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576000000x_2x_5^4+165888000x_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
5 6 7 7 2
o13 = (map(R,R,{-x + 3x + x , x , -x + -x + x , x }), ideal (-x + 3x x
2 1 2 4 1 5 1 6 2 3 2 2 1 1 2
-----------------------------------------------------------------------
3 391 2 2 7 3 5 2 2 6 2
+ x x + 1, 3x x + ---x x + -x x + -x x x + 3x x x + -x x x +
1 4 1 2 60 1 2 2 1 2 2 1 2 3 1 2 3 5 1 2 4
-----------------------------------------------------------------------
7 2
-x x x + x x x x + 1), {x , x })
6 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
7 9 10 1 11 2
o16 = (map(R,R,{-x + -x + x , x , --x + -x + x , x }), ideal (--x +
4 1 5 2 4 1 3 1 2 2 3 2 4 1
-----------------------------------------------------------------------
9 35 3 55 2 2 9 3 7 2 9 2
-x x + x x + 1, --x x + --x x + --x x + -x x x + -x x x +
5 1 2 1 4 6 1 2 8 1 2 10 1 2 4 1 2 3 5 1 2 3
-----------------------------------------------------------------------
10 2 1 2
--x x x + -x x x + x x x x + 1), {x , x })
3 1 2 4 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
--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,{- 3x + x , x , - 3x + 3x + x , x }), ideal (x - 3x x +
2 4 1 1 2 3 2 1 1 2
-----------------------------------------------------------------------
2 2 3 2 2 2
x x + 1, 9x x - 9x x - 3x x x - 3x x x + 3x x x + x x x x + 1),
1 4 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.