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
1 1 10 3 2
o3 = (map(R,R,{-x + 4x + x , x , -x + --x + x , x }), ideal (-x + 4x x
2 1 2 4 1 8 1 9 2 3 2 2 1 1 2
------------------------------------------------------------------------
1 3 19 2 2 40 3 1 2 2 1 2
+ x x + 1, --x x + --x x + --x x + -x x x + 4x x x + -x x x +
1 4 16 1 2 18 1 2 9 1 2 2 1 2 3 1 2 3 8 1 2 4
------------------------------------------------------------------------
10 2
--x x x + x x x x + 1), {x , x })
9 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
1 7 5 2 7
o6 = (map(R,R,{-x + -x + x , x , -x + -x + x , x + -x + x , x }), ideal
2 1 8 2 5 1 2 1 3 2 4 1 9 2 3 2
------------------------------------------------------------------------
1 2 7 3 1 3 21 2 2 3 2 147 3 21 2
(-x + -x x + x x - x , -x x + --x x + -x x x + ---x x + --x x x
2 1 8 1 2 1 5 2 8 1 2 32 1 2 4 1 2 5 128 1 2 8 1 2 5
------------------------------------------------------------------------
3 2 343 4 147 3 21 2 2 3
+ -x x x + ---x + ---x x + --x x + x x ), {x , x , x })
2 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} | 32768x_1x_2x_5^6-75264x_2^9x_5-16807x_2^9+
{-9} | 9604x_1x_2^2x_5^3-24576x_1x_2x_5^5+10976x_
{-9} | 7909306972x_1x_2^3+20239392768x_1x_2^2x_5^
{-3} | 4x_1^2+7x_1x_2+8x_1x_5-8x_2^3
------------------------------------------------------------------------
43008x_2^8x_5^2+19208x_2^8x_5-16384x_2^7x_5^3-21952x_2^7x_5^2+25088x_2
1x_2x_5^4+56448x_2^9-32256x_2^8x_5-4802x_2^8+12288x_2^7x_5^2+10976x_2^
2+18078415936x_1x_2^2x_5+77309411328x_1x_2x_5^5-17263755264x_1x_2x_5^4
------------------------------------------------------------------------
^6x_5^3-28672x_2^5x_5^4+32768x_2^4x_5^5+57344x_2^2x_5^6+65536x_2x
7x_5-18816x_2^6x_5^2+21504x_2^5x_5^3-24576x_2^4x_5^4+10976x_2^4x_
+15420489728x_1x_2x_5^3+10330523392x_1x_2x_5^2-177570054144x_2^9+
------------------------------------------------------------------------
_5^7
5^3+16807x_2^3x_5^3-43008x_2^2x_5^5+38416x_2^2x_5^4-49152x_2x_5^6+21952x
101468602368x_2^8x_5+22658678784x_2^8-38654705664x_2^7x_5^2-43159388160x
------------------------------------------------------------------------
_2x_5^5
_2^7x_5+3855122432x_2^7+59190018048x_2^6x_5^2-13217562624x_2^6x_5-
------------------------------------------------------------------------
5903156224x_2^6-67645734912x_2^5x_5^3+15105785856x_2^5x_5^2+6746464256x_
------------------------------------------------------------------------
2^5x_5+9039207968x_2^5+77309411328x_2^4x_5^4-17263755264x_2^4x_5^3+
------------------------------------------------------------------------
15420489728x_2^4x_5^2+10330523392x_2^4x_5+13841287201x_2^4+35418937344x_
------------------------------------------------------------------------
2^3x_5^2+47455841832x_2^3x_5+135291469824x_2^2x_5^5-30211571712x_2^2x_5^
------------------------------------------------------------------------
4+67464642560x_2^2x_5^3+54235247808x_2^2x_5^2+154618822656x_2x_5^6-
------------------------------------------------------------------------
|
|
|
34527510528x_2x_5^5+30840979456x_2x_5^4+20661046784x_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
6 7 1 11 2 7
o13 = (map(R,R,{-x + -x + x , x , -x + x + x , x }), ideal (--x + -x x
5 1 6 2 4 1 3 1 2 3 2 5 1 6 1 2
-----------------------------------------------------------------------
2 3 143 2 2 7 3 6 2 7 2 1 2
+ x x + 1, -x x + ---x x + -x x + -x x x + -x x x + -x x x +
1 4 5 1 2 90 1 2 6 1 2 5 1 2 3 6 1 2 3 3 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
3 1 1 4 11 2 1
o16 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (--x + -x x
8 1 5 2 4 1 9 1 9 2 3 2 8 1 5 1 2
-----------------------------------------------------------------------
1 3 17 2 2 4 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 24 1 2 90 1 2 45 1 2 8 1 2 3 5 1 2 3 9 1 2 4
-----------------------------------------------------------------------
4 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
2
o19 = (map(R,R,{2x + x + x , x , - x + x + x , x }), ideal (3x + x x +
1 2 4 1 1 2 3 2 1 1 2
-----------------------------------------------------------------------
3 2 2 3 2 2 2 2
x x + 1, - 2x x + x x + x x + 2x x x + x x x - x x x + x 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.