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
5 7 1 9 2 7
o3 = (map(R,R,{-x + -x + x , x , x + -x + x , x }), ideal (-x + -x x +
4 1 8 2 4 1 1 4 2 3 2 4 1 8 1 2
------------------------------------------------------------------------
5 3 19 2 2 7 3 5 2 7 2 2 1 2
x x + 1, -x x + --x x + --x x + -x x x + -x x x + x x x + -x x x
1 4 4 1 2 16 1 2 32 1 2 4 1 2 3 8 1 2 3 1 2 4 4 1 2 4
------------------------------------------------------------------------
+ x x x x + 1), {x , x })
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
8 8 3 8 5
o6 = (map(R,R,{4x + -x + x , x , -x + -x + x , -x + -x + x , x }),
1 3 2 5 1 5 1 2 2 4 5 1 3 2 3 2
------------------------------------------------------------------------
2 8 3 3 2 2 2 256 3
ideal (4x + -x x + x x - x , 64x x + 128x x + 48x x x + ---x x +
1 3 1 2 1 5 2 1 2 1 2 1 2 5 3 1 2
------------------------------------------------------------------------
2 2 512 4 64 3 2 2 3
64x x x + 12x x x + ---x + --x x + 8x x + x x ), {x , x , x })
1 2 5 1 2 5 27 2 3 2 5 2 5 2 5 5 4 3
o6 : Sequence
|
i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 972x_1x_2x_5^6-165888x_2^9x_5-131072x_2^9+31104x_2^
{-9} | 12288x_1x_2^2x_5^3-2916x_1x_2x_5^5+4608x_1x_2x_5^4+
{-9} | 805306368x_1x_2^3+191102976x_1x_2^2x_5^2+603979776x
{-3} | 12x_1^2+8x_1x_2+3x_1x_5-3x_2^3
------------------------------------------------------------------------
8x_5^2+49152x_2^8x_5-3888x_2^7x_5^3-18432x_2^7x_5^2+6912x_2^6x_5^3-
497664x_2^9-93312x_2^8x_5-49152x_2^8+11664x_2^7x_5^2+36864x_2^7x_5-
_1x_2^2x_5+19131876x_1x_2x_5^5-15116544x_1x_2x_5^4+47775744x_1x_2x_
------------------------------------------------------------------------
2592x_2^5x_5^4+972x_2^4x_5^5+648x_2^2x_5^6+243x_2x_5^7
20736x_2^6x_5^2+7776x_2^5x_5^3-2916x_2^4x_5^4+4608x_2^4x_5^3+8192x_2^3x_
5^3+113246208x_1x_2x_5^2-3265173504x_2^9+612220032x_2^8x_5+483729408x_2^
------------------------------------------------------------------------
5^3-1944x_2^2x_5^5+6144x_2^2x_5^4-729x_2x_5^6+1152x_2x_5^5
8-76527504x_2^7x_5^2-302330880x_2^7x_5+95551488x_2^7+136048896x_2^6x_5^2
------------------------------------------------------------------------
-107495424x_2^6x_5-169869312x_2^6-51018336x_2^5x_5^3+40310784x_2^5x_5^2+
------------------------------------------------------------------------
63700992x_2^5x_5+301989888x_2^5+19131876x_2^4x_5^4-15116544x_2^4x_5^3+
------------------------------------------------------------------------
47775744x_2^4x_5^2+113246208x_2^4x_5+536870912x_2^4+127401984x_2^3x_5^2+
------------------------------------------------------------------------
603979776x_2^3x_5+12754584x_2^2x_5^5-10077696x_2^2x_5^4+79626240x_2^2x_5
------------------------------------------------------------------------
^3+226492416x_2^2x_5^2+4782969x_2x_5^6-3779136x_2x_5^5+11943936x_2x_5^4+
------------------------------------------------------------------------
|
|
|
28311552x_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,{a + 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
10 7 19 2
o13 = (map(R,R,{--x + 2x + x , x , -x + x + x , x }), ideal (--x + 2x x
9 1 2 4 1 9 1 2 3 2 9 1 1 2
-----------------------------------------------------------------------
70 3 8 2 2 3 10 2 2 7 2
+ x x + 1, --x x + -x x + 2x x + --x x x + 2x x x + -x x x +
1 4 81 1 2 3 1 2 1 2 9 1 2 3 1 2 3 9 1 2 4
-----------------------------------------------------------------------
2
x x x + x x x x + 1), {x , x })
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
1 1 8 2 6 2 1
o16 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (-x + -x x
5 1 6 2 4 1 7 1 9 2 3 2 5 1 6 1 2
-----------------------------------------------------------------------
8 3 74 2 2 1 3 1 2 1 2 8 2
+ x x + 1, --x x + ---x x + --x x + -x x x + -x x x + -x x x +
1 4 35 1 2 315 1 2 27 1 2 5 1 2 3 6 1 2 3 7 1 2 4
-----------------------------------------------------------------------
2 2
-x x x + x x x x + 1), {x , x })
9 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
--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
2
o19 = (map(R,R,{2x + x , x , 2x + x , x }), ideal (x + 2x x + x x + 1,
2 4 1 2 3 2 1 1 2 1 4
-----------------------------------------------------------------------
3 2 2
4x x + 2x x x + 2x x x + x x x x + 1), {x , x })
1 2 1 2 3 1 2 4 1 2 3 4 4 3
o19 : Sequence
|
This symbol is provided by the package NoetherNormalization.