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 9 6 2 9
o3 = (map(R,R,{-x + -x + x , x , x + 3x + x , x }), ideal (-x + -x x +
5 1 5 2 4 1 1 2 3 2 5 1 5 1 2
------------------------------------------------------------------------
1 3 12 2 2 27 3 1 2 9 2 2 2
x x + 1, -x x + --x x + --x x + -x x x + -x x x + x x x + 3x x x
1 4 5 1 2 5 1 2 5 1 2 5 1 2 3 5 1 2 3 1 2 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
1 6 5 1 4
o6 = (map(R,R,{-x + -x + x , x , -x + -x + x , -x + 3x + x , x }),
6 1 7 2 5 1 6 1 4 2 4 9 1 2 3 2
------------------------------------------------------------------------
1 2 6 3 1 3 1 2 2 1 2 18 3
ideal (-x + -x x + x x - x , ---x x + --x x + --x x x + --x x +
6 1 7 1 2 1 5 2 216 1 2 14 1 2 12 1 2 5 49 1 2
------------------------------------------------------------------------
6 2 1 2 216 4 108 3 18 2 2 3
-x x x + -x x x + ---x + ---x x + --x x + x x ), {x , x , x })
7 1 2 5 2 1 2 5 343 2 49 2 5 7 2 5 2 5 5 4 3
o6 : Sequence
|
i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 100842x_1x_2x_5^6-74088x_2^9x_5-46656x
{-9} | 108864x_1x_2^2x_5^3-100842x_1x_2x_5^5+
{-9} | 182849716224x_1x_2^3+169375836672x_1x_
{-3} | 7x_1^2+36x_1x_2+42x_1x_5-42x_2^3
------------------------------------------------------------------------
_2^9+43218x_2^8x_5^2+54432x_2^8x_5-16807x_2^7x_5^3-
127008x_1x_2x_5^4+74088x_2^9-43218x_2^8x_5-18144x_2
2^2x_5^2+426649337856x_1x_2^2x_5+83047723206x_1x_2x
------------------------------------------------------------------------
63504x_2^7x_5^2+74088x_2^6x_5^3-86436x_2^5x_5^4+100842x_2^4x_5^5+518616x
^8+16807x_2^7x_5^2+42336x_2^7x_5-74088x_2^6x_5^2+86436x_2^5x_5^3-100842x
_5^5-52298274672x_1x_2x_5^4+131736761856x_1x_2x_5^3+248878780416x_1x_2x_
------------------------------------------------------------------------
_2^2x_5^6+605052x_2x_5^7
_2^4x_5^4+127008x_2^4x_5^3+559872x_2^3x_5^3-518616x_2^2x_5^5+1306368x_2^
5^2-61014653784x_2^9+35591881374x_2^8x_5+22413546288x_2^8-13841287201x_2
------------------------------------------------------------------------
2x_5^4-605052x_2x_5^6+762048x_2x_5^5
^7x_5^2-43581895560x_2^7x_5+10978063488x_2^7+61014653784x_2^6x_5^2-
------------------------------------------------------------------------
38423222208x_2^6x_5-48393096192x_2^6-71183762748x_2^5x_5^3+44827092576x_
------------------------------------------------------------------------
2^5x_5^2+56458612224x_2^5x_5+213324668928x_2^5+83047723206x_2^4x_5^4-
------------------------------------------------------------------------
52298274672x_2^4x_5^3+131736761856x_2^4x_5^2+248878780416x_2^4x_5+
------------------------------------------------------------------------
940369969152x_2^4+871075731456x_2^3x_5^2+3291294892032x_2^3x_5+
------------------------------------------------------------------------
427102576488x_2^2x_5^5-268962555456x_2^2x_5^4+1693758366720x_2^2x_5^3+
------------------------------------------------------------------------
3839844040704x_2^2x_5^2+498286339236x_2x_5^6-313789648032x_2x_5^5+
------------------------------------------------------------------------
|
|
|
790420571136x_2x_5^4+1493272682496x_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
1 7 7 8 2 7
o13 = (map(R,R,{-x + -x + x , x , 2x + -x + x , x }), ideal (-x + -x x
7 1 8 2 4 1 1 6 2 3 2 7 1 8 1 2
-----------------------------------------------------------------------
2 3 23 2 2 49 3 1 2 7 2 2
+ x x + 1, -x x + --x x + --x x + -x x x + -x x x + 2x x x +
1 4 7 1 2 12 1 2 48 1 2 7 1 2 3 8 1 2 3 1 2 4
-----------------------------------------------------------------------
7 2
-x x x + x x x x + 1), {x , x })
6 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 1 3 4 2 1
o16 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (-x + -x x
3 1 3 2 4 1 8 1 5 2 3 2 3 1 3 1 2
-----------------------------------------------------------------------
1 3 29 2 2 1 3 1 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 2 120 1 2 5 1 2 3 1 2 3 3 1 2 3 8 1 2 4
-----------------------------------------------------------------------
3 2
-x x x + x x x x + 1), {x , x })
5 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
--trying with basis element limit: 20
2
o19 = (map(R,R,{- 3x - 2x + x , x , - 3x + 2x + x , x }), ideal (- 2x -
1 2 4 1 1 2 3 2 1
-----------------------------------------------------------------------
3 3 2 2 2 2
2x x + x x + 1, 9x x - 4x x - 3x x x - 2x x x - 3x x x + 2x x x
1 2 1 4 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.