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
5 5 2 9 2 5
o3 = (map(R,R,{-x + -x + x , x , 2x + -x + x , x }), ideal (-x + -x x +
4 1 8 2 4 1 1 3 2 3 2 4 1 8 1 2
------------------------------------------------------------------------
5 3 25 2 2 5 3 5 2 5 2 2
x x + 1, -x x + --x x + --x x + -x x x + -x x x + 2x x x +
1 4 2 1 2 12 1 2 12 1 2 4 1 2 3 8 1 2 3 1 2 4
------------------------------------------------------------------------
2 2
-x x x + x x x x + 1), {x , x })
3 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
5 9 2 1
o6 = (map(R,R,{-x + -x + x , x , -x + x + x , -x + 5x + x , x }), ideal
2 1 4 2 5 1 3 1 2 4 3 1 2 3 2
------------------------------------------------------------------------
5 2 9 3 125 3 675 2 2 75 2 1215 3
(-x + -x x + x x - x , ---x x + ---x x + --x x x + ----x x +
2 1 4 1 2 1 5 2 8 1 2 16 1 2 4 1 2 5 32 1 2
------------------------------------------------------------------------
135 2 15 2 729 4 243 3 27 2 2 3
---x x x + --x x x + ---x + ---x x + --x x + x x ), {x , x , x })
4 1 2 5 2 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} | 5120x_1x_2x_5^6-388800x_2^9x_5-295245x_2^9+86400x_2^8x_5^2
{-9} | 21870x_1x_2^2x_5^3-6400x_1x_2x_5^5+9720x_1x_2x_5^4+486000x
{-9} | 34867844010x_1x_2^3+10203667200x_1x_2^2x_5^2+30993639120x_
{-3} | 10x_1^2+9x_1x_2+4x_1x_5-4x_2^3
------------------------------------------------------------------------
+131220x_2^8x_5-12800x_2^7x_5^3-58320x_2^7x_5^2+25920x_2^6x_5^3-11520x_2
_2^9-108000x_2^8x_5-54675x_2^8+16000x_2^7x_5^2+48600x_2^7x_5-32400x_2^6x
1x_2^2x_5+1310720000x_1x_2x_5^5-995328000x_1x_2x_5^4+3023308800x_1x_2x_5
------------------------------------------------------------------------
^5x_5^4+5120x_2^4x_5^5+4608x_2^2x_5^6+2048x_2x_5^7
_5^2+14400x_2^5x_5^3-6400x_2^4x_5^4+9720x_2^4x_5^3+19683x_2^3x
^3+6887475360x_1x_2x_5^2-99532800000x_2^9+22118400000x_2^8x_5+
------------------------------------------------------------------------
_5^3-5760x_2^2x_5^5+17496x_2^2x_5^4-2560x_2x_5^6+3888x_2x_5^5
16796160000x_2^8-3276800000x_2^7x_5^2-12441600000x_2^7x_5+3779136000x_2^
------------------------------------------------------------------------
7+6635520000x_2^6x_5^2-5038848000x_2^6x_5-7652750400x_2^6-2949120000x_2^
------------------------------------------------------------------------
5x_5^3+2239488000x_2^5x_5^2+3401222400x_2^5x_5+15496819560x_2^5+
------------------------------------------------------------------------
1310720000x_2^4x_5^4-995328000x_2^4x_5^3+3023308800x_2^4x_5^2+
------------------------------------------------------------------------
6887475360x_2^4x_5+31381059609x_2^4+9183300480x_2^3x_5^2+41841412812x_2^
------------------------------------------------------------------------
3x_5+1179648000x_2^2x_5^5-895795200x_2^2x_5^4+6802444800x_2^2x_5^3+
------------------------------------------------------------------------
18596183472x_2^2x_5^2+524288000x_2x_5^6-398131200x_2x_5^5+1209323520x_2x
------------------------------------------------------------------------
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_5^4+2754990144x_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
5 4 1 9 9 2 4
o13 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (-x + -x x
4 1 9 2 4 1 5 1 2 2 3 2 4 1 9 1 2
-----------------------------------------------------------------------
1 3 2057 2 2 3 5 2 4 2 1 2
+ x x + 1, -x x + ----x x + 2x x + -x x x + -x x x + -x x x +
1 4 4 1 2 360 1 2 1 2 4 1 2 3 9 1 2 3 5 1 2 4
-----------------------------------------------------------------------
9 2
-x x x + x x x x + 1), {x , x })
2 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
7 9 1 16 2 9
o16 = (map(R,R,{-x + -x + x , x , -x + 2x + x , x }), ideal (--x + -x x
9 1 8 2 4 1 5 1 2 3 2 9 1 8 1 2
-----------------------------------------------------------------------
7 3 641 2 2 9 3 7 2 9 2 1 2
+ x x + 1, --x x + ---x x + -x x + -x x x + -x x x + -x x x +
1 4 45 1 2 360 1 2 4 1 2 9 1 2 3 8 1 2 3 5 1 2 4
-----------------------------------------------------------------------
2
2x x x + x x x x + 1), {x , x })
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
--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: 3
--trying with basis element limit: 5
--trying with basis element limit: 20
2
o19 = (map(R,R,{- 6x - 3x + x , x , - x + x , x }), ideal (- 5x - 3x x +
1 2 4 1 2 3 2 1 1 2
-----------------------------------------------------------------------
2 2 3 2 2 2
x x + 1, 6x x + 3x x - 6x x x - 3x x x - x x x + x x x x + 1),
1 4 1 2 1 2 1 2 3 1 2 3 1 2 4 1 2 3 4
-----------------------------------------------------------------------
{x , x })
4 3
o19 : Sequence
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This symbol is provided by the package NoetherNormalization.