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
10 1 9 17 2 1
o3 = (map(R,R,{--x + -x + x , x , -x + 6x + x , x }), ideal (--x + -x x
7 1 2 2 4 1 7 1 2 3 2 7 1 2 1 2
------------------------------------------------------------------------
90 3 129 2 2 3 10 2 1 2 9 2
+ x x + 1, --x x + ---x x + 3x x + --x x x + -x x x + -x x x +
1 4 49 1 2 14 1 2 1 2 7 1 2 3 2 1 2 3 7 1 2 4
------------------------------------------------------------------------
2
6x x x + x x x x + 1), {x , x })
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
5 3 3 1 10
o6 = (map(R,R,{-x + -x + x , x , -x + 2x + x , --x + --x + x , x }),
3 1 8 2 5 1 5 1 2 4 10 1 7 2 3 2
------------------------------------------------------------------------
5 2 3 3 125 3 25 2 2 25 2 45 3
ideal (-x + -x x + x x - x , ---x x + --x x + --x x x + --x x +
3 1 8 1 2 1 5 2 27 1 2 8 1 2 3 1 2 5 64 1 2
------------------------------------------------------------------------
15 2 2 27 4 27 3 9 2 2 3
--x x x + 5x x x + ---x + --x x + -x x + x x ), {x , x , x })
4 1 2 5 1 2 5 512 2 64 2 5 8 2 5 2 5 5 4 3
o6 : Sequence
|
i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 491520x_1x_2x_5^6-691200x_2^9x_5-3645x_2^9+921600x_2^
{-9} | 9720x_1x_2^2x_5^3-2457600x_1x_2x_5^5+25920x_1x_2x_5^4
{-9} | 63772920x_1x_2^3+16124313600x_1x_2^2x_5^2+340122240x_
{-3} | 40x_1^2+9x_1x_2+24x_1x_5-24x_2^3
------------------------------------------------------------------------
8x_5^2+9720x_2^8x_5-819200x_2^7x_5^3-25920x_2^7x_5^2+69120x
+3456000x_2^9-4608000x_2^8x_5-16200x_2^8+4096000x_2^7x_5^2+
1x_2^2x_5+257698037760000x_1x_2x_5^5-1358954496000x_1x_2x_5
------------------------------------------------------------------------
_2^6x_5^3-184320x_2^5x_5^4+491520x_2^4x_5^5+
86400x_2^7x_5-345600x_2^6x_5^2+921600x_2^5x_
^4+28665446400x_1x_2x_5^3+453496320x_1x_2x_5
------------------------------------------------------------------------
110592x_2^2x_5^6+294912x_2x_5^7
5^3-2457600x_2^4x_5^4+25920x_2^4x_5^3+2187x_2^3x_5^3-552960x_2^2x_5
^2-362387865600000x_2^9+483183820800000x_2^8x_5+2548039680000x_2^8-
------------------------------------------------------------------------
^5+11664x_2^2x_5^4-1474560x_2x_5^6+15552x_2x_5^5
429496729600000x_2^7x_5^2-11324620800000x_2^7x_5+23887872000x_2^7+
------------------------------------------------------------------------
36238786560000x_2^6x_5^2-191102976000x_2^6x_5-2015539200x_2^6-
------------------------------------------------------------------------
96636764160000x_2^5x_5^3+509607936000x_2^5x_5^2+5374771200x_2^5x_5+
------------------------------------------------------------------------
170061120x_2^5+257698037760000x_2^4x_5^4-1358954496000x_2^4x_5^3+
------------------------------------------------------------------------
28665446400x_2^4x_5^2+453496320x_2^4x_5+14348907x_2^4+3627970560x_2^3x_5
------------------------------------------------------------------------
^2+114791256x_2^3x_5+57982058496000x_2^2x_5^5-305764761600x_2^2x_5^4+
------------------------------------------------------------------------
16124313600x_2^2x_5^3+306110016x_2^2x_5^2+154618822656000x_2x_5^6-
------------------------------------------------------------------------
|
|
|
815372697600x_2x_5^5+17199267840x_2x_5^4+272097792x_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 1 1 7 10 2 1
o13 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (--x + -x x
7 1 2 2 4 1 2 1 4 2 3 2 7 1 2 1 2
-----------------------------------------------------------------------
3 3 2 2 7 3 3 2 1 2 1 2
+ x x + 1, --x x + x x + -x x + -x x x + -x x x + -x x x +
1 4 14 1 2 1 2 8 1 2 7 1 2 3 2 1 2 3 2 1 2 4
-----------------------------------------------------------------------
7 2
-x x x + x x x x + 1), {x , x })
4 1 2 4 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
4 1 4 7 2
o16 = (map(R,R,{-x + x + x , x , -x + -x + x , x }), ideal (-x + x x +
3 1 2 4 1 4 1 5 2 3 2 3 1 1 2
-----------------------------------------------------------------------
1 3 79 2 2 4 3 4 2 2 1 2 4 2
x x + 1, -x x + --x x + -x x + -x x x + x x x + -x x x + -x x x
1 4 3 1 2 60 1 2 5 1 2 3 1 2 3 1 2 3 4 1 2 4 5 1 2 4
-----------------------------------------------------------------------
+ x x x x + 1), {x , x })
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,{x - 2x + x , x , x - 2x + x , x }), ideal (2x - 2x x +
1 2 4 1 1 2 3 2 1 1 2
-----------------------------------------------------------------------
3 2 2 3 2 2 2 2
x x + 1, x x - 4x x + 4x x + x x x - 2x x x + x x x - 2x x x +
1 4 1 2 1 2 1 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
o19 : Sequence
|
This symbol is provided by the package NoetherNormalization.