Description
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
9 9 1 1 13 2
o3 = (map (R, R, {-x + -x + x , x , -x + -x + x , x }), ideal (--x +
4 1 2 2 4 1 2 1 2 2 3 2 4 1
------------------------------------------------------------------------
9 9 3 27 2 2 9 3 9 2 9 2 1 2
-x x + x x + 1, -x x + --x x + -x x + -x x x + -x x x + -x x x +
2 1 2 1 4 8 1 2 8 1 2 4 1 2 4 1 2 3 2 1 2 3 2 1 2 4
------------------------------------------------------------------------
1 2
-x x x + x x x x + 1), {x , x })
2 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 7 3 7 3
o6 = (map (R, R, {-x + x + x , x , -x + -x + x , -x + -x + x , x }),
4 1 2 5 1 4 1 4 2 4 9 1 2 2 3 2
------------------------------------------------------------------------
3 2 3 27 3 27 2 2 27 2 9 3
ideal (-x + x x + x x - x , --x x + --x x + --x x x + -x x +
4 1 1 2 1 5 2 64 1 2 16 1 2 16 1 2 5 4 1 2
------------------------------------------------------------------------
9 2 9 2 4 3 2 2 3
-x x x + -x x x + x + 3x x + 3x x + x x ), {x , x , x })
2 1 2 5 4 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} | 12x_1x_2x_5^6-54x_2^9x_5-12x_2^9+27x_2^8x_5^2+12x_2^8x_5-9x_2^7x
{-9} | 48x_1x_2^2x_5^3-108x_1x_2x_5^5+48x_1x_2x_5^4+486x_2^9-243x_2^8x_
{-9} | 384x_1x_2^3+864x_1x_2^2x_5^2+768x_1x_2^2x_5+2916x_1x_2x_5^5-648x
{-3} | 3x_1^2+4x_1x_2+4x_1x_5-4x_2^3
------------------------------------------------------------------------
_5^3-12x_2^7x_5^2+12x_2^6x_5^3-12x_2^5x_5^4+12x_2^4x_5^5+16x_2
5-36x_2^8+81x_2^7x_5^2+72x_2^7x_5-108x_2^6x_5^2+108x_2^5x_5^3-
_1x_2x_5^4+576x_1x_2x_5^3+384x_1x_2x_5^2-13122x_2^9+6561x_2^8x
------------------------------------------------------------------------
^2x_5^6+16x_2x_5^7
108x_2^4x_5^4+48x_2^4x_5^3+64x_2^3x_5^3-144x_2^2x_5^5+128x_2^2x_5^4-144x
_5+1458x_2^8-2187x_2^7x_5^2-2430x_2^7x_5+216x_2^7+2916x_2^6x_5^2-648x_2^
------------------------------------------------------------------------
_2x_5^6+64x_2x_5^5
6x_5-288x_2^6-2916x_2^5x_5^3+648x_2^5x_5^2+288x_2^5x_5+384x_2^5+2916x_2^
------------------------------------------------------------------------
4x_5^4-648x_2^4x_5^3+576x_2^4x_5^2+384x_2^4x_5+512x_2^4+1152x_2^3x_5^2+
------------------------------------------------------------------------
1536x_2^3x_5+3888x_2^2x_5^5-864x_2^2x_5^4+1920x_2^2x_5^3+1536x_2^2x_5^2+
------------------------------------------------------------------------
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3888x_2x_5^6-864x_2x_5^5+768x_2x_5^4+512x_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
|
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 3 7 2
o13 = (map (R, R, {7x + --x + x , x , -x + -x + x , x }), ideal (8x +
1 10 2 4 1 7 1 3 2 3 2 1
-----------------------------------------------------------------------
7 3 499 2 2 49 3 2 7 2
--x x + x x + 1, 3x x + ---x x + --x x + 7x x x + --x x x +
10 1 2 1 4 1 2 30 1 2 30 1 2 1 2 3 10 1 2 3
-----------------------------------------------------------------------
3 2 7 2
-x x x + -x x x + x x x x + 1), {x , x })
7 1 2 4 3 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
2 5 6 5 2
o16 = (map (R, R, {-x + -x + x , x , x + -x + x , x }), ideal (-x +
3 1 2 2 4 1 1 7 2 3 2 3 1
-----------------------------------------------------------------------
5 2 3 43 2 2 15 3 2 2 5 2 2
-x x + x x + 1, -x x + --x x + --x x + -x x x + -x x x + x x x
2 1 2 1 4 3 1 2 14 1 2 7 1 2 3 1 2 3 2 1 2 3 1 2 4
-----------------------------------------------------------------------
6 2
+ -x x x + x x x x + 1), {x , x })
7 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 (3x - x x + x x
1 2 4 1 2 3 2 1 1 2 1 4
-----------------------------------------------------------------------
2 2 3 2 2 2
+ 1, 2x x - x x + 2x x x - x x x + x x x + x x x x + 1), {x ,
1 2 1 2 1 2 3 1 2 3 1 2 4 1 2 3 4 4
-----------------------------------------------------------------------
x })
3
o19 : Sequence
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This symbol is provided by the package NoetherNormalization.