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
6 2 1 11 2 2
o3 = (map(R,R,{-x + -x + x , x , x + -x + x , x }), ideal (--x + -x x +
5 1 5 2 4 1 1 4 2 3 2 5 1 5 1 2
------------------------------------------------------------------------
6 3 7 2 2 1 3 6 2 2 2 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 5 1 2 10 1 2 10 1 2 5 1 2 3 5 1 2 3 1 2 4 4 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
2 3 7
o6 = (map(R,R,{5x + -x + x , x , x + --x + x , -x + 3x + x , x }),
1 5 2 5 1 1 10 2 4 4 1 2 3 2
------------------------------------------------------------------------
2 2 3 3 2 2 2 12 3
ideal (5x + -x x + x x - x , 125x x + 30x x + 75x x x + --x x +
1 5 1 2 1 5 2 1 2 1 2 1 2 5 5 1 2
------------------------------------------------------------------------
2 2 8 4 12 3 6 2 2 3
12x x x + 15x x x + ---x + --x x + -x x + x x ), {x , x , x })
1 2 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} | 3125x_1x_2x_5^6-15000x_2^9x_5-32x_2^9+18750x_2^8x_5
{-9} | 400x_1x_2^2x_5^3-234375x_1x_2x_5^5+1000x_1x_2x_5^4+
{-9} | 25600x_1x_2^3+15000000x_1x_2^2x_5^2+128000x_1x_2^2x
{-3} | 25x_1^2+2x_1x_2+5x_1x_5-5x_2^3
------------------------------------------------------------------------
^2+80x_2^8x_5-15625x_2^7x_5^3-200x_2^7x_5^2+500x_2^6x_5^3-1250x_2^5x_5
1125000x_2^9-1406250x_2^8x_5-2000x_2^8+1171875x_2^7x_5^2+10000x_2^7x_5
_5+1373291015625x_1x_2x_5^5-2929687500x_1x_2x_5^4+25000000x_1x_2x_5^3+
------------------------------------------------------------------------
^4+3125x_2^4x_5^5+250x_2^2x_5^6+625x_2x_5^7
-37500x_2^6x_5^2+93750x_2^5x_5^3-234375x_2^4x_5^4+1000x_2^4x_5^3+32x_2^
160000x_1x_2x_5^2-6591796875000x_2^9+8239746093750x_2^8x_5+17578125000x
------------------------------------------------------------------------
3x_5^3-18750x_2^2x_5^5+160x_2^2x_5^4-46875x_2x_5^6+200x_2x_5^5
_2^8-6866455078125x_2^7x_5^2-73242187500x_2^7x_5+62500000x_2^7+
------------------------------------------------------------------------
219726562500x_2^6x_5^2-468750000x_2^6x_5-2000000x_2^6-549316406250x_2^5x
------------------------------------------------------------------------
_5^3+1171875000x_2^5x_5^2+5000000x_2^5x_5+64000x_2^5+1373291015625x_2^4x
------------------------------------------------------------------------
_5^4-2929687500x_2^4x_5^3+25000000x_2^4x_5^2+160000x_2^4x_5+2048x_2^4+
------------------------------------------------------------------------
1200000x_2^3x_5^2+15360x_2^3x_5+109863281250x_2^2x_5^5-234375000x_2^2x_5
------------------------------------------------------------------------
^4+5000000x_2^2x_5^3+38400x_2^2x_5^2+274658203125x_2x_5^6-585937500x_2x_
------------------------------------------------------------------------
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5^5+5000000x_2x_5^4+32000x_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 10 8 2 4
o13 = (map(R,R,{4x + -x + x , x , --x + -x + x , x }), ideal (5x + -x x
1 7 2 4 1 3 1 3 2 3 2 1 7 1 2
-----------------------------------------------------------------------
40 3 88 2 2 32 3 2 4 2 10 2
+ x x + 1, --x x + --x x + --x x + 4x x x + -x x x + --x x x +
1 4 3 1 2 7 1 2 21 1 2 1 2 3 7 1 2 3 3 1 2 4
-----------------------------------------------------------------------
8 2
-x x x + x x x x + 1), {x , x })
3 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
2 1 2 2
o16 = (map(R,R,{5x + -x + x , x , 6x + --x + x , x }), ideal (6x + -x x
1 3 2 4 1 1 10 2 3 2 1 3 1 2
-----------------------------------------------------------------------
3 9 2 2 1 3 2 2 2 2
+ x x + 1, 30x x + -x x + --x x + 5x x x + -x x x + 6x x x +
1 4 1 2 2 1 2 15 1 2 1 2 3 3 1 2 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
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,{- 4x - x + x , x , - 4x - 6x + x , x }), ideal (- 3x -
1 2 4 1 1 2 3 2 1
-----------------------------------------------------------------------
3 2 2 3 2 2 2
x x + x x + 1, 16x x + 28x x + 6x x - 4x x x - x x x - 4x x x -
1 2 1 4 1 2 1 2 1 2 1 2 3 1 2 3 1 2 4
-----------------------------------------------------------------------
2
6x x x + x x x x + 1), {x , x })
1 2 4 1 2 3 4 4 3
o19 : Sequence
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This symbol is provided by the package NoetherNormalization.