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 1 2 1
o3 = (map(R,R,{x + -x + x , x , -x + -x + x , x }), ideal (2x + -x x +
1 6 2 4 1 2 1 2 2 3 2 1 6 1 2
------------------------------------------------------------------------
3 3 3 2 2 1 3 2 1 2 3 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 2 1 2 4 1 2 12 1 2 1 2 3 6 1 2 3 2 1 2 4 2 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
9 1 3 1
o6 = (map(R,R,{-x + -x + x , x , x + -x + x , 9x + -x + x , x }), ideal
4 1 4 2 5 1 1 5 2 4 1 2 2 3 2
------------------------------------------------------------------------
9 2 1 3 729 3 243 2 2 243 2 27 3
(-x + -x x + x x - x , ---x x + ---x x + ---x x x + --x x +
4 1 4 1 2 1 5 2 64 1 2 64 1 2 16 1 2 5 64 1 2
------------------------------------------------------------------------
27 2 27 2 1 4 3 3 3 2 2 3
--x x x + --x x x + --x + --x x + -x x + x x ), {x , x , x })
8 1 2 5 4 1 2 5 64 2 16 2 5 4 2 5 2 5 5 4 3
o6 : Sequence
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i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 9216x_1x_2x_5^6-7776x_2^9x_5-9x_2^9+15552x_2^8x_5^2+36x_2^
{-9} | 36x_1x_2^2x_5^3-62208x_1x_2x_5^5+144x_1x_2x_5^4+52488x_2^9
{-9} | 9x_1x_2^3+15552x_1x_2^2x_5^2+72x_1x_2^2x_5+7739670528x_1x_
{-3} | 9x_1^2+x_1x_2+4x_1x_5-4x_2^3
------------------------------------------------------------------------
8x_5-20736x_2^7x_5^3-144x_2^7x_5^2+576x_2^6x_5^3
-104976x_2^8x_5-81x_2^8+139968x_2^7x_5^2+648x_2^
2x_5^5-8957952x_1x_2x_5^4+41472x_1x_2x_5^3+144x_
------------------------------------------------------------------------
-2304x_2^5x_5^4+9216x_2^4x_5^5+1024x_2^2x_5^6+4096x_2x_5^7
7x_5-3888x_2^6x_5^2+15552x_2^5x_5^3-62208x_2^4x_5^4+144x_2^
1x_2x_5^2-6530347008x_2^9+13060694016x_2^8x_5+15116544x_2^8
------------------------------------------------------------------------
4x_5^3+4x_2^3x_5^3-6912x_2^2x_5^5+32x_2^2x_5^4-27648x_2x_5^6+64x_2x_5^5
-17414258688x_2^7x_5^2-100776960x_2^7x_5+46656x_2^7+483729408x_2^6x_5^2-
------------------------------------------------------------------------
559872x_2^6x_5-1296x_2^6-1934917632x_2^5x_5^3+2239488x_2^5x_5^2+5184x_2^
------------------------------------------------------------------------
5x_5+36x_2^5+7739670528x_2^4x_5^4-8957952x_2^4x_5^3+41472x_2^4x_5^2+144x
------------------------------------------------------------------------
_2^4x_5+x_2^4+1728x_2^3x_5^2+12x_2^3x_5+859963392x_2^2x_5^5-995328x_2^2x
------------------------------------------------------------------------
_5^4+11520x_2^2x_5^3+48x_2^2x_5^2+3439853568x_2x_5^6-3981312x_2x_5^5+
------------------------------------------------------------------------
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18432x_2x_5^4+64x_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 1 2
o13 = (map(R,R,{x + 8x + x , x , -x + -x + x , x }), ideal (2x + 8x x +
1 2 4 1 3 1 3 2 3 2 1 1 2
-----------------------------------------------------------------------
5 3 41 2 2 8 3 2 2 5 2 1 2
x x + 1, -x x + --x x + -x x + x x x + 8x x x + -x x x + -x x x
1 4 3 1 2 3 1 2 3 1 2 1 2 3 1 2 3 3 1 2 4 3 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
1 8 4 4 2 8
o16 = (map(R,R,{-x + -x + x , x , -x + 3x + x , x }), ideal (-x + -x x
3 1 5 2 4 1 3 1 2 3 2 3 1 5 1 2
-----------------------------------------------------------------------
4 3 47 2 2 24 3 1 2 8 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 15 1 2 5 1 2 3 1 2 3 5 1 2 3 3 1 2 4
-----------------------------------------------------------------------
2
3x x x + x x x x + 1), {x , x })
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,{- 3x + 2x + x , x , - 2x + x , x }), ideal (- 2x + 2x x
1 2 4 1 2 3 2 1 1 2
-----------------------------------------------------------------------
2 2 3 2 2 2
+ x x + 1, 6x x - 4x x - 3x x x + 2x x x - 2x x x + x x x x +
1 4 1 2 1 2 1 2 3 1 2 3 1 2 4 1 2 3 4
-----------------------------------------------------------------------
1), {x , x })
4 3
o19 : Sequence
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This symbol is provided by the package NoetherNormalization.