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
83 28 2
o3 = (map(R,R,{2x + --x + x , x , 2x + --x + x , x }), ideal (3x +
1 8 2 4 1 1 5 2 3 2 1
------------------------------------------------------------------------
83 3 639 2 2 581 3 2 83 2
--x x + x x + 1, 4x x + ---x x + ---x x + 2x x x + --x x x +
8 1 2 1 4 1 2 20 1 2 10 1 2 1 2 3 8 1 2 3
------------------------------------------------------------------------
2 28 2
2x x x + --x x x + x x x x + 1), {x , x })
1 2 4 5 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
1 9 4 3
o6 = (map(R,R,{-x + 10x + x , x , -x + 4x + x , -x + --x + x , x }),
2 1 2 5 1 7 1 2 4 9 1 10 2 3 2
------------------------------------------------------------------------
1 2 3 1 3 15 2 2 3 2 3
ideal (-x + 10x x + x x - x , -x x + --x x + -x x x + 150x x +
2 1 1 2 1 5 2 8 1 2 2 1 2 4 1 2 5 1 2
------------------------------------------------------------------------
2 3 2 4 3 2 2 3
30x x x + -x x x + 1000x + 300x x + 30x x + x x ), {x , x , x })
1 2 5 2 1 2 5 2 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} | 2x_1x_2x_5^6-600x_2^9x_5-200000x_2^9+30x_2^8x_5^2+20000x_2^8x_5
{-9} | 40000x_1x_2^2x_5^3-6x_1x_2x_5^5+4000x_1x_2x_5^4+1800x_2^9-90x_2
{-9} | 800000000000x_1x_2^3+120000000x_1x_2^2x_5^2+160000000000x_1x_2^
{-3} | x_1^2+20x_1x_2+2x_1x_5-2x_2^3
------------------------------------------------------------------------
-x_2^7x_5^3-2000x_2^7x_5^2+200x_2^6x_5^3-20x_2^5x_5^4+2x_2^4x_5^5+40x_2^
^8x_5-20000x_2^8+3x_2^7x_5^2+4000x_2^7x_5-600x_2^6x_5^2+60x_2^5x_5^3-6x_
2x_5+18x_1x_2x_5^5-6000x_1x_2x_5^4+8000000x_1x_2x_5^3+8000000000x_1x_2x_
------------------------------------------------------------------------
2x_5^6+4x_2x_5^7
2^4x_5^4+4000x_2^4x_5^3+800000x_2^3x_5^3-120x_2^2x_5^5+160000x_2^2x_5^4-
5^2-5400x_2^9+270x_2^8x_5+90000x_2^8-9x_2^7x_5^2-15000x_2^7x_5+2000000x_
------------------------------------------------------------------------
12x_2x_5^6+8000x_2x_5^5
2^7+1800x_2^6x_5^2-600000x_2^6x_5-400000000x_2^6-180x_2^5x_5^3+60000x_2^
------------------------------------------------------------------------
5x_5^2+40000000x_2^5x_5+80000000000x_2^5+18x_2^4x_5^4-6000x_2^4x_5^3+
------------------------------------------------------------------------
8000000x_2^4x_5^2+8000000000x_2^4x_5+16000000000000x_2^4+2400000000x_2^
------------------------------------------------------------------------
3x_5^2+4800000000000x_2^3x_5+360x_2^2x_5^5-120000x_2^2x_5^4+400000000x_2
------------------------------------------------------------------------
^2x_5^3+480000000000x_2^2x_5^2+36x_2x_5^6-12000x_2x_5^5+16000000x_2x_5^4
------------------------------------------------------------------------
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+16000000000x_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
3 1 1 2
o13 = (map(R,R,{10x + -x + x , x , -x + -x + x , x }), ideal (11x +
1 5 2 4 1 6 1 7 2 3 2 1
-----------------------------------------------------------------------
3 5 3 107 2 2 3 3 2 3 2
-x x + x x + 1, -x x + ---x x + --x x + 10x x x + -x x x +
5 1 2 1 4 3 1 2 70 1 2 35 1 2 1 2 3 5 1 2 3
-----------------------------------------------------------------------
1 2 1 2
-x x x + -x x x + x x x x + 1), {x , x })
6 1 2 4 7 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 4 4 6 8 2 4
o16 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (-x + -x x
3 1 9 2 4 1 3 1 5 2 3 2 3 1 9 1 2
-----------------------------------------------------------------------
20 3 70 2 2 8 3 5 2 4 2 4 2
+ x x + 1, --x x + --x x + --x x + -x x x + -x x x + -x x x +
1 4 9 1 2 27 1 2 15 1 2 3 1 2 3 9 1 2 3 3 1 2 4
-----------------------------------------------------------------------
6 2
-x x x + x x x x + 1), {x , x })
5 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
2
o19 = (map(R,R,{- 5x + x , x , - x + x , x }), ideal (x - 5x x + x x +
2 4 1 2 3 2 1 1 2 1 4
-----------------------------------------------------------------------
3 2 2
1, 5x x - 5x 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.