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 9 4 6 2 9
o3 = (map(R,R,{-x + -x + x , x , -x + 4x + x , x }), ideal (-x + -x x +
5 1 8 2 4 1 5 1 2 3 2 5 1 8 1 2
------------------------------------------------------------------------
4 3 17 2 2 9 3 1 2 9 2 4 2
x x + 1, --x x + --x x + -x x + -x x x + -x x x + -x x x +
1 4 25 1 2 10 1 2 2 1 2 5 1 2 3 8 1 2 3 5 1 2 4
------------------------------------------------------------------------
2
4x 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
9 4 7 7 4
o6 = (map(R,R,{-x + -x + x , x , 4x + -x + x , -x + -x + x , x }),
7 1 5 2 5 1 1 5 2 4 8 1 3 2 3 2
------------------------------------------------------------------------
9 2 4 3 729 3 972 2 2 243 2 432 3
ideal (-x + -x x + x x - x , ---x x + ---x x + ---x x x + ---x x
7 1 5 1 2 1 5 2 343 1 2 245 1 2 49 1 2 5 175 1 2
------------------------------------------------------------------------
216 2 27 2 64 4 48 3 12 2 2 3
+ ---x x x + --x x x + ---x + --x x + --x x + x x ), {x , x , x })
35 1 2 5 7 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} | 196875x_1x_2x_5^6-972000x_2^9x_5-64512x
{-9} | 564480x_1x_2^2x_5^3-5315625x_1x_2x_5^5+
{-9} | 226586787840x_1x_2^3+2133734400000x_1x_
{-3} | 45x_1^2+28x_1x_2+35x_1x_5-35x_2^3
------------------------------------------------------------------------
_2^9+607500x_2^8x_5^2+80640x_2^8x_5-253125x_2^7x_5^3-100800x
705600x_1x_2x_5^4+26244000x_2^9-16402500x_2^8x_5-725760x_2^8
2^2x_5^2+566466969600x_1x_2^2x_5+100913818359375x_1x_2x_5^5-
------------------------------------------------------------------------
_2^7x_5^2+126000x_2^6x_5^3-157500x_2^5x_5^4+196875x_2^4x_5^5+122500x_2^
+6834375x_2^7x_5^2+1814400x_2^7x_5-3402000x_2^6x_5^2+4252500x_2^5x_5^3-
6697687500000x_1x_2x_5^4+1778112000000x_1x_2x_5^3+354041856000x_1x_2x_5
------------------------------------------------------------------------
2x_5^6+153125x_2x_5^7
5315625x_2^4x_5^4+705600x_2^4x_5^3+351232x_2^3x_5^3-3307500x_2^2x_5^
^2-498225937500000x_2^9+311391210937500x_2^8x_5+20667150000000x_2^8-
------------------------------------------------------------------------
5+878080x_2^2x_5^4-4134375x_2x_5^6+548800x_2x_5^5
129746337890625x_2^7x_5^2-43056562500000x_2^7x_5+1143072000000x_2^7+
------------------------------------------------------------------------
64584843750000x_2^6x_5^2-4286520000000x_2^6x_5-568995840000x_2^6-
------------------------------------------------------------------------
80731054687500x_2^5x_5^3+5358150000000x_2^5x_5^2+711244800000x_2^5x_5+
------------------------------------------------------------------------
283233484800x_2^5+100913818359375x_2^4x_5^4-6697687500000x_2^4x_5^3+
------------------------------------------------------------------------
1778112000000x_2^4x_5^2+354041856000x_2^4x_5+140987334656x_2^4+
------------------------------------------------------------------------
1327656960000x_2^3x_5^2+528702504960x_2^3x_5+62790820312500x_2^2x_5^5-
------------------------------------------------------------------------
4167450000000x_2^2x_5^4+2765952000000x_2^2x_5^3+660878131200x_2^2x_5^2+
------------------------------------------------------------------------
78488525390625x_2x_5^6-5209312500000x_2x_5^5+1382976000000x_2x_5^4+
------------------------------------------------------------------------
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275365888000x_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
9 1 2 9
o13 = (map(R,R,{2x + -x + x , x , -x + 2x + x , x }), ideal (3x + -x x
1 5 2 4 1 4 1 2 3 2 1 5 1 2
-----------------------------------------------------------------------
1 3 89 2 2 18 3 2 9 2 1 2
+ x x + 1, -x x + --x x + --x x + 2x x x + -x x x + -x x x +
1 4 2 1 2 20 1 2 5 1 2 1 2 3 5 1 2 3 4 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 5 9 2 5
o16 = (map(R,R,{8x + -x + x , x , -x + -x + x , x }), ideal (9x + -x x
1 7 2 4 1 2 1 8 2 3 2 1 7 1 2
-----------------------------------------------------------------------
3 151 2 2 45 3 2 5 2 5 2
+ x x + 1, 20x x + ---x x + --x x + 8x x x + -x x x + -x x x +
1 4 1 2 14 1 2 56 1 2 1 2 3 7 1 2 3 2 1 2 4
-----------------------------------------------------------------------
9 2
-x x x + x x x x + 1), {x , x })
8 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
2
o19 = (map(R,R,{2x + x , x , x - x + x , x }), ideal (x + 2x x + x x +
2 4 1 1 2 3 2 1 1 2 1 4
-----------------------------------------------------------------------
2 2 3 2 2 2
1, 2x x - 2x x + 2x x x + x x x - x x x + x x x x + 1), {x , x })
1 2 1 2 1 2 3 1 2 4 1 2 4 1 2 3 4 4 3
o19 : Sequence
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This symbol is provided by the package NoetherNormalization.