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];
|
i2 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);
o2 : Ideal of R
|
i3 : (f,J,X) = noetherNormalization I
2 1 4 5 5 2 1
o3 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (-x + -x x +
3 1 2 2 4 1 9 1 6 2 3 2 3 1 2 1 2
------------------------------------------------------------------------
8 3 7 2 2 5 3 2 2 1 2 4 2
x x + 1, --x x + -x x + --x x + -x x x + -x x x + -x x x +
1 4 27 1 2 9 1 2 12 1 2 3 1 2 3 2 1 2 3 9 1 2 4
------------------------------------------------------------------------
5 2
-x x x + x x x x + 1), {x , x })
6 1 2 4 1 2 3 4 4 3
o3 : Sequence
|
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];
|
i5 : I = ideal(x_2*x_1-x_5^3, x_5*x_1^3);
o5 : Ideal of R
|
i6 : (f,J,X) = noetherNormalization I
3 5 5 4 5 2
o6 = (map(R,R,{-x + -x + x , x , -x + -x + x , -x + -x + x , x }),
5 1 2 2 5 1 8 1 3 2 4 4 1 5 2 3 2
------------------------------------------------------------------------
3 2 5 3 27 3 27 2 2 27 2 45 3
ideal (-x + -x x + x x - x , ---x x + --x x + --x x x + --x x +
5 1 2 1 2 1 5 2 125 1 2 10 1 2 25 1 2 5 4 1 2
------------------------------------------------------------------------
2 9 2 125 4 75 3 15 2 2 3
9x x x + -x x x + ---x + --x x + --x x + x x ), {x , x , x })
1 2 5 5 1 2 5 8 2 4 2 5 2 2 5 2 5 5 4 3
o6 : Sequence
|
i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 480x_1x_2x_5^6-10800x_2^9x_5-46875x_2^9+2160x_2^8x_5^2+18750x_2
{-9} | 93750x_1x_2^2x_5^3-4320x_1x_2x_5^5+37500x_1x_2x_5^4+97200x_2^9-
{-9} | 183105468750x_1x_2^3+8437500000x_1x_2^2x_5^2+146484375000x_1x_2
{-3} | 6x_1^2+25x_1x_2+10x_1x_5-10x_2^3
------------------------------------------------------------------------
^8x_5-288x_2^7x_5^3-7500x_2^7x_5^2+3000x_2^6x_5^3-1200x_2^5x_5^4+480x
19440x_2^8x_5-56250x_2^8+2592x_2^7x_5^2+45000x_2^7x_5-27000x_2^6x_5^2
^2x_5+29859840x_1x_2x_5^5-129600000x_1x_2x_5^4+2250000000x_1x_2x_5^3+
------------------------------------------------------------------------
_2^4x_5^5+2000x_2^2x_5^6+800x_2x_5^7
+10800x_2^5x_5^3-4320x_2^4x_5^4+37500x_2^4x_5^3+390625x_2^3x_5^3-18000x
29296875000x_1x_2x_5^2-671846400x_2^9+134369280x_2^8x_5+583200000x_2^8-
------------------------------------------------------------------------
_2^2x_5^5+312500x_2^2x_5^4-7200x_2x_5^6+62500x_2x_5^5
17915904x_2^7x_5^2-388800000x_2^7x_5+675000000x_2^7+186624000x_2^6x_5^2-
------------------------------------------------------------------------
810000000x_2^6x_5-7031250000x_2^6-74649600x_2^5x_5^3+324000000x_2^5x_5^2
------------------------------------------------------------------------
+2812500000x_2^5x_5+73242187500x_2^5+29859840x_2^4x_5^4-129600000x_2^4x_
------------------------------------------------------------------------
5^3+2250000000x_2^4x_5^2+29296875000x_2^4x_5+762939453125x_2^4+
------------------------------------------------------------------------
35156250000x_2^3x_5^2+915527343750x_2^3x_5+124416000x_2^2x_5^5-
------------------------------------------------------------------------
540000000x_2^2x_5^4+23437500000x_2^2x_5^3+366210937500x_2^2x_5^2+
------------------------------------------------------------------------
49766400x_2x_5^6-216000000x_2x_5^5+3750000000x_2x_5^4+48828125000x_2x_5^
------------------------------------------------------------------------
|
|
|
3 |
|
5 1
o7 : Matrix R <--- R
|
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];
|
i9 : I = ideal(a^2*b+a*b^2+1);
o9 : Ideal of R
|
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];
|
i12 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);
o12 : Ideal of R
|
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 5 2
o13 = (map(R,R,{-x + 8x + x , x , 3x + x + x , x }), ideal (-x + 8x x +
2 1 2 4 1 1 2 3 2 2 1 1 2
-----------------------------------------------------------------------
9 3 51 2 2 3 3 2 2 2 2
x x + 1, -x x + --x x + 8x x + -x x x + 8x x x + 3x x x + x x x
1 4 2 1 2 2 1 2 1 2 2 1 2 3 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];
|
i15 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);
o15 : Ideal of R
|
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
10 9 2
o16 = (map(R,R,{10x + x + x , x , --x + -x + x , x }), ideal (11x + x x
1 2 4 1 3 1 4 2 3 2 1 1 2
-----------------------------------------------------------------------
100 3 155 2 2 9 3 2 2 10 2
+ x x + 1, ---x x + ---x x + -x x + 10x x x + x x x + --x x x +
1 4 3 1 2 6 1 2 4 1 2 1 2 3 1 2 3 3 1 2 4
-----------------------------------------------------------------------
9 2
-x x x + x x x x + 1), {x , x })
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];
|
i18 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);
o18 : Ideal of R
|
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 - 2x + x , x , - 2x - x + x , x }), ideal (- x -
1 2 4 1 1 2 3 2 1
-----------------------------------------------------------------------
3 2 2 3 2 2 2
2x x + x x + 1, 4x x + 6x x + 2x x - 2x 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
|
This symbol is provided by the package NoetherNormalization.