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 1 2 2
o3 = (map(R,R,{x + -x + x , x , -x + -x + x , x }), ideal (2x + -x x +
1 5 2 4 1 8 1 3 2 3 2 1 5 1 2
------------------------------------------------------------------------
7 3 41 2 2 2 3 2 2 2 7 2 1 2
x x + 1, -x x + --x x + --x x + x x x + -x x x + -x x x + -x x x
1 4 8 1 2 60 1 2 15 1 2 1 2 3 5 1 2 3 8 1 2 4 3 1 2 4
------------------------------------------------------------------------
+ x x x x + 1), {x , x })
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 8 8 1
o6 = (map(R,R,{-x + -x + x , x , -x + -x + x , 9x + -x + x , x }),
4 1 2 2 5 1 5 1 3 2 4 1 2 2 3 2
------------------------------------------------------------------------
5 2 7 3 125 3 525 2 2 75 2 735 3
ideal (-x + -x x + x x - x , ---x x + ---x x + --x x x + ---x x +
4 1 2 1 2 1 5 2 64 1 2 32 1 2 16 1 2 5 16 1 2
------------------------------------------------------------------------
105 2 15 2 343 4 147 3 21 2 2 3
---x x x + --x x x + ---x + ---x x + --x x + x x ), {x , x , x })
4 1 2 5 4 1 2 5 8 2 4 2 5 2 2 5 2 5 5 4 3
o6 : Sequence
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i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 160x_1x_2x_5^6-14700x_2^9x_5-84035x_2^9+2100x_2^8x_5^2+24010x_2^
{-9} | 48020x_1x_2^2x_5^3-1200x_1x_2x_5^5+13720x_1x_2x_5^4+110250x_2^9-
{-9} | 9886633715x_1x_2^3+247062900x_1x_2^2x_5^2+5649504980x_1x_2^2x_5+
{-3} | 5x_1^2+14x_1x_2+4x_1x_5-4x_2^3
------------------------------------------------------------------------
8x_5-200x_2^7x_5^3-6860x_2^7x_5^2+1960x_2^6x_5^3-560x_2^5x_5^4+160x_2^
15750x_2^8x_5-60025x_2^8+1500x_2^7x_5^2+34300x_2^7x_5-14700x_2^6x_5^2+
360000x_1x_2x_5^5-2058000x_1x_2x_5^4+47059600x_1x_2x_5^3+807072140x_1x
------------------------------------------------------------------------
4x_5^5+448x_2^2x_5^6+128x_2x_5^7
4200x_2^5x_5^3-1200x_2^4x_5^4+13720x_2^4x_5^3+134456x_2^3x_5^3-3360x_
_2x_5^2-33075000x_2^9+4725000x_2^8x_5+27011250x_2^8-450000x_2^7x_5^2-
------------------------------------------------------------------------
2^2x_5^5+76832x_2^2x_5^4-960x_2x_5^6+10976x_2x_5^5
12862500x_2^7x_5+29412250x_2^7+4410000x_2^6x_5^2-25210500x_2^6x_5-
------------------------------------------------------------------------
288240050x_2^6-1260000x_2^5x_5^3+7203000x_2^5x_5^2+82354300x_2^5x_5+
------------------------------------------------------------------------
2824752490x_2^5+360000x_2^4x_5^4-2058000x_2^4x_5^3+47059600x_2^4x_5^2+
------------------------------------------------------------------------
807072140x_2^4x_5+27682574402x_2^4+691776120x_2^3x_5^2+23727920916x_2^3x
------------------------------------------------------------------------
_5+1008000x_2^2x_5^5-5762400x_2^2x_5^4+329417200x_2^2x_5^3+6779405976x_2
------------------------------------------------------------------------
^2x_5^2+288000x_2x_5^6-1646400x_2x_5^5+37647680x_2x_5^4+645657712x_2x_5^
------------------------------------------------------------------------
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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 1 5 2 1
o13 = (map(R,R,{-x + -x + x , x , 9x + 2x + x , x }), ideal (-x + -x x
3 1 7 2 4 1 1 2 3 2 3 1 7 1 2
-----------------------------------------------------------------------
3 55 2 2 2 3 2 2 1 2 2
+ x x + 1, 6x x + --x x + -x x + -x x x + -x x x + 9x x x +
1 4 1 2 21 1 2 7 1 2 3 1 2 3 7 1 2 3 1 2 4
-----------------------------------------------------------------------
2
2x x x + x x x x + 1), {x , x })
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 7 14 2
o16 = (map(R,R,{-x + 10x + x , x , --x + 2x + x , x }), ideal (--x +
9 1 2 4 1 10 1 2 3 2 9 1
-----------------------------------------------------------------------
7 3 73 2 2 3 5 2 2
10x x + x x + 1, --x x + --x x + 20x x + -x x x + 10x x x +
1 2 1 4 18 1 2 9 1 2 1 2 9 1 2 3 1 2 3
-----------------------------------------------------------------------
7 2 2
--x x x + 2x x x + x x x x + 1), {x , x })
10 1 2 4 1 2 4 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
2
o19 = (map(R,R,{x + x , x , - x + x , x }), ideal (x + x x + x x + 1, -
2 4 1 2 3 2 1 1 2 1 4
-----------------------------------------------------------------------
3 2 2
x x + x x x - x x x + x x x x + 1), {x , x })
1 2 1 2 3 1 2 4 1 2 3 4 4 3
o19 : Sequence
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This symbol is provided by the package NoetherNormalization.