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
10 9 1 5 17 2 9
o3 = (map(R,R,{--x + -x + x , x , -x + -x + x , x }), ideal (--x + -x x
7 1 7 2 4 1 2 1 3 2 3 2 7 1 7 1 2
------------------------------------------------------------------------
5 3 127 2 2 15 3 10 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 7 1 2 42 1 2 7 1 2 7 1 2 3 7 1 2 3 2 1 2 4
------------------------------------------------------------------------
5 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
3 1 3 7
o6 = (map(R,R,{-x + x + x , x , -x + 9x + x , -x + -x + x , x }), ideal
8 1 2 5 1 5 1 2 4 2 1 9 2 3 2
------------------------------------------------------------------------
3 2 3 27 3 27 2 2 27 2 9 3 9 2
(-x + x x + x x - x , ---x x + --x x + --x x x + -x x + -x x x +
8 1 1 2 1 5 2 512 1 2 64 1 2 64 1 2 5 8 1 2 4 1 2 5
------------------------------------------------------------------------
9 2 4 3 2 2 3
-x x x + x + 3x x + 3x x + x x ), {x , x , x })
8 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} | 24x_1x_2x_5^6-54x_2^9x_5-24x_2^9+27x_2^8x_5^2+24x_2^8x_5-9x_2^
{-9} | 192x_1x_2^2x_5^3-216x_1x_2x_5^5+192x_1x_2x_5^4+486x_2^9-243x_2
{-9} | 6144x_1x_2^3+6912x_1x_2^2x_5^2+12288x_1x_2^2x_5+5832x_1x_2x_5^
{-3} | 3x_1^2+8x_1x_2+8x_1x_5-8x_2^3
------------------------------------------------------------------------
7x_5^3-24x_2^7x_5^2+24x_2^6x_5^3-24x_2^5x_5^4+24x_2^4x_5^5+64x_2^2x_5^6+
^8x_5-72x_2^8+81x_2^7x_5^2+144x_2^7x_5-216x_2^6x_5^2+216x_2^5x_5^3-216x_
5-2592x_1x_2x_5^4+4608x_1x_2x_5^3+6144x_1x_2x_5^2-13122x_2^9+6561x_2^8x_
------------------------------------------------------------------------
64x_2x_5^7
2^4x_5^4+192x_2^4x_5^3+512x_2^3x_5^3-576x_2^2x_5^5+1024x_2^2x_5^4-576x_
5+2916x_2^8-2187x_2^7x_5^2-4860x_2^7x_5+864x_2^7+5832x_2^6x_5^2-2592x_2
------------------------------------------------------------------------
2x_5^6+512x_2x_5^5
^6x_5-2304x_2^6-5832x_2^5x_5^3+2592x_2^5x_5^2+2304x_2^5x_5+6144x_2^5+
------------------------------------------------------------------------
5832x_2^4x_5^4-2592x_2^4x_5^3+4608x_2^4x_5^2+6144x_2^4x_5+16384x_2^4+
------------------------------------------------------------------------
18432x_2^3x_5^2+49152x_2^3x_5+15552x_2^2x_5^5-6912x_2^2x_5^4+30720x_2^2x
------------------------------------------------------------------------
_5^3+49152x_2^2x_5^2+15552x_2x_5^6-6912x_2x_5^5+12288x_2x_5^4+16384x_2x_
------------------------------------------------------------------------
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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
7 6 2 8 9 2 6
o13 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (-x + -x x
2 1 5 2 4 1 3 1 7 2 3 2 2 1 5 1 2
-----------------------------------------------------------------------
7 3 24 2 2 48 3 7 2 6 2 2 2
+ x x + 1, -x x + --x x + --x x + -x x x + -x x x + -x x x +
1 4 3 1 2 5 1 2 35 1 2 2 1 2 3 5 1 2 3 3 1 2 4
-----------------------------------------------------------------------
8 2
-x x x + x x x x + 1), {x , x })
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
9 5 10 2 9
o16 = (map(R,R,{x + -x + x , x , -x + --x + x , x }), ideal (2x + -x x
1 8 2 4 1 4 1 3 2 3 2 1 8 1 2
-----------------------------------------------------------------------
5 3 455 2 2 15 3 2 9 2 5 2
+ x x + 1, -x x + ---x x + --x x + x x x + -x x x + -x x x +
1 4 4 1 2 96 1 2 4 1 2 1 2 3 8 1 2 3 4 1 2 4
-----------------------------------------------------------------------
10 2
--x x x + x x x x + 1), {x , x })
3 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
--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,{- 3x + x + x , x , - 4x + 4x + x , x }), ideal (- 2x +
1 2 4 1 1 2 3 2 1
-----------------------------------------------------------------------
3 2 2 3 2 2 2
x x + x x + 1, 12x x - 16x x + 4x x - 3x x x + x x x - 4x x x +
1 2 1 4 1 2 1 2 1 2 1 2 3 1 2 3 1 2 4
-----------------------------------------------------------------------
2
4x 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.