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 8 5 3 2 8
o3 = (map(R,R,{-x + -x + x , x , -x + x + x , x }), ideal (-x + -x x +
2 1 9 2 4 1 2 1 2 3 2 2 1 9 1 2
------------------------------------------------------------------------
5 3 49 2 2 8 3 1 2 8 2 5 2 2
x x + 1, -x x + --x x + -x x + -x x x + -x x x + -x x x + x x x
1 4 4 1 2 18 1 2 9 1 2 2 1 2 3 9 1 2 3 2 1 2 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
7 9 9 1 9
o6 = (map(R,R,{4x + -x + x , x , -x + -x + x , -x + -x + x , x }),
1 8 2 5 1 8 1 4 2 4 9 1 4 2 3 2
------------------------------------------------------------------------
2 7 3 3 2 2 2 147 3
ideal (4x + -x x + x x - x , 64x x + 42x x + 48x x x + ---x x +
1 8 1 2 1 5 2 1 2 1 2 1 2 5 16 1 2
------------------------------------------------------------------------
2 2 343 4 147 3 21 2 2 3
21x x x + 12x x x + ---x + ---x x + --x x + x x ), {x , x , x })
1 2 5 1 2 5 512 2 64 2 5 8 2 5 2 5 5 4 3
o6 : Sequence
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i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 32768x_1x_2x_5^6-602112x_2^9x_5-16807x
{-9} | 76832x_1x_2^2x_5^3-1572864x_1x_2x_5^5+
{-9} | 63274455776x_1x_2^3+1295321137152x_1x_
{-3} | 32x_1^2+7x_1x_2+8x_1x_5-8x_2^3
------------------------------------------------------------------------
_2^9+344064x_2^8x_5^2+19208x_2^8x_5-131072x_2^7x_5^3-21952x_
87808x_1x_2x_5^4+28901376x_2^9-16515072x_2^8x_5-307328x_2^8+
2^2x_5^2+144627327488x_1x_2^2x_5+316659348799488x_1x_2x_5^5-
------------------------------------------------------------------------
2^7x_5^2+25088x_2^6x_5^3-28672x_2^5x_5^4+32768x_2^4x_5^5+7168x_2^2x_5
6291456x_2^7x_5^2+702464x_2^7x_5-1204224x_2^6x_5^2+1376256x_2^5x_5^3-
8839042695168x_1x_2x_5^4+986911342592x_1x_2x_5^3+82644187136x_1x_2x_5
------------------------------------------------------------------------
^6+8192x_2x_5^7
1572864x_2^4x_5^4+87808x_2^4x_5^3+16807x_2^3x_5^3-344064x_2^2x_5^5+
^2-5818615534190592x_2^9+3324923162394624x_2^8x_5+92809948299264x_2
------------------------------------------------------------------------
38416x_2^2x_5^4-393216x_2x_5^6+21952x_2x_5^5
^8-1266637395197952x_2^7x_5^2-176780853903360x_2^7x_5+1973822685184x_2^7
------------------------------------------------------------------------
+242442313924608x_2^6x_5^2-6767392063488x_2^6x_5-377801998336x_2^6-
------------------------------------------------------------------------
277076930199552x_2^5x_5^3+7734162358272x_2^5x_5^2+431773712384x_2^5x_5+
------------------------------------------------------------------------
72313663744x_2^5+316659348799488x_2^4x_5^4-8839042695168x_2^4x_5^3+
------------------------------------------------------------------------
986911342592x_2^4x_5^2+82644187136x_2^4x_5+13841287201x_2^4+
------------------------------------------------------------------------
283351498752x_2^3x_5^2+47455841832x_2^3x_5+69269232549888x_2^2x_5^5-
------------------------------------------------------------------------
1933540589568x_2^2x_5^4+539717140480x_2^2x_5^3+54235247808x_2^2x_5^2+
------------------------------------------------------------------------
79164837199872x_2x_5^6-2209760673792x_2x_5^5+246727835648x_2x_5^4+
------------------------------------------------------------------------
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20661046784x_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,{a + 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 2 3
o13 = (map(R,R,{2x + -x + x , x , x + x + x , x }), ideal (3x + -x x +
1 4 2 4 1 1 2 3 2 1 4 1 2
-----------------------------------------------------------------------
3 11 2 2 3 3 2 3 2 2 2
x x + 1, 2x x + --x x + -x x + 2x x x + -x x x + x x x + x x x
1 4 1 2 4 1 2 4 1 2 1 2 3 4 1 2 3 1 2 4 1 2 4
-----------------------------------------------------------------------
+ x x x x + 1), {x , x })
1 2 3 4 4 3
o13 : Sequence
|
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
3 11 2
o16 = (map(R,R,{-x + 10x + x , x , x + 5x + x , x }), ideal (--x +
8 1 2 4 1 1 2 3 2 8 1
-----------------------------------------------------------------------
3 3 95 2 2 3 3 2 2
10x x + x x + 1, -x x + --x x + 50x x + -x x x + 10x x x +
1 2 1 4 8 1 2 8 1 2 1 2 8 1 2 3 1 2 3
-----------------------------------------------------------------------
2 2
x x x + 5x x x + x x x x + 1), {x , x })
1 2 4 1 2 4 1 2 3 4 4 3
o16 : Sequence
|
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 - 2x + x , x , 2x + x + x , x }), ideal (- 3x -
1 2 4 1 1 2 3 2 1
-----------------------------------------------------------------------
3 2 2 3 2 2 2
2x x + x x + 1, - 8x x - 8x x - 2x x - 4x x x - 2x x x + 2x x x
1 2 1 4 1 2 1 2 1 2 1 2 3 1 2 3 1 2 4
-----------------------------------------------------------------------
2
+ x 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.