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 2 2
o3 = (map(R,R,{4x + -x + x , x , -x + 5x + x , x }), ideal (5x + -x x +
1 3 2 4 1 7 1 2 3 2 1 3 1 2
------------------------------------------------------------------------
4 3 422 2 2 10 3 2 2 2 1 2
x x + 1, -x x + ---x x + --x x + 4x x x + -x x x + -x x x +
1 4 7 1 2 21 1 2 3 1 2 1 2 3 3 1 2 3 7 1 2 4
------------------------------------------------------------------------
2
5x 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
1 2 3 3 1 5
o6 = (map(R,R,{-x + -x + x , x , -x + -x + x , -x + -x + x , x }),
2 1 3 2 5 1 4 1 5 2 4 2 1 4 2 3 2
------------------------------------------------------------------------
1 2 2 3 1 3 1 2 2 3 2 2 3
ideal (-x + -x x + x x - x , -x x + -x x + -x x x + -x x +
2 1 3 1 2 1 5 2 8 1 2 2 1 2 4 1 2 5 3 1 2
------------------------------------------------------------------------
2 3 2 8 4 4 3 2 2 3
2x x x + -x x x + --x + -x x + 2x x + x x ), {x , x , x })
1 2 5 2 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} | 486x_1x_2x_5^6-648x_2^9x_5-64x_2^9+486x_2^8x_5^2+96x_2^8x_5-243x
{-9} | 192x_1x_2^2x_5^3-1458x_1x_2x_5^5+288x_1x_2x_5^4+1944x_2^9-1458x_
{-9} | 49152x_1x_2^3+373248x_1x_2^2x_5^2+147456x_1x_2^2x_5+9565938x_1x_
{-3} | 3x_1^2+4x_1x_2+6x_1x_5-6x_2^3
------------------------------------------------------------------------
_2^7x_5^3-144x_2^7x_5^2+216x_2^6x_5^3-324x_2^5x_5^4+486x_2^4x
2^8x_5-96x_2^8+729x_2^7x_5^2+288x_2^7x_5-648x_2^6x_5^2+972x_2
2x_5^5-944784x_1x_2x_5^4+373248x_1x_2x_5^3+110592x_1x_2x_5^2-
------------------------------------------------------------------------
_5^5+648x_2^2x_5^6+972x_2x_5^7
^5x_5^3-1458x_2^4x_5^4+288x_2^4x_5^3+256x_2^3x_5^3-1944x_2^2x_5^5+768x_
12754584x_2^9+9565938x_2^8x_5+944784x_2^8-4782969x_2^7x_5^2-2361960x_2^
------------------------------------------------------------------------
2^2x_5^4-2916x_2x_5^6+576x_2x_5^5
7x_5+93312x_2^7+4251528x_2^6x_5^2-419904x_2^6x_5-82944x_2^6-6377292x_2^
------------------------------------------------------------------------
5x_5^3+629856x_2^5x_5^2+124416x_2^5x_5+73728x_2^5+9565938x_2^4x_5^4-
------------------------------------------------------------------------
944784x_2^4x_5^3+373248x_2^4x_5^2+110592x_2^4x_5+65536x_2^4+497664x_2^3x
------------------------------------------------------------------------
_5^2+294912x_2^3x_5+12754584x_2^2x_5^5-1259712x_2^2x_5^4+1244160x_2^2x_5
------------------------------------------------------------------------
^3+442368x_2^2x_5^2+19131876x_2x_5^6-1889568x_2x_5^5+746496x_2x_5^4+
------------------------------------------------------------------------
|
|
|
221184x_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 2 2 3 2
o13 = (map(R,R,{-x + 3x + x , x , -x + -x + x , x }), ideal (-x + 3x x
2 1 2 4 1 3 1 5 2 3 2 2 1 1 2
-----------------------------------------------------------------------
1 3 11 2 2 6 3 1 2 2 2 2
+ x x + 1, -x x + --x x + -x x + -x x x + 3x x x + -x x x +
1 4 3 1 2 5 1 2 5 1 2 2 1 2 3 1 2 3 3 1 2 4
-----------------------------------------------------------------------
2 2
-x x x + x x x x + 1), {x , x })
5 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
7 7 5 5 11 2 7
o16 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (--x + -x x
4 1 6 2 4 1 4 1 2 2 3 2 4 1 6 1 2
-----------------------------------------------------------------------
35 3 35 2 2 35 3 7 2 7 2 5 2
+ x x + 1, --x x + --x x + --x x + -x x x + -x x x + -x x x +
1 4 16 1 2 6 1 2 12 1 2 4 1 2 3 6 1 2 3 4 1 2 4
-----------------------------------------------------------------------
5 2
-x x x + x x x x + 1), {x , x })
2 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
--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: 3
--trying with basis element limit: 5
--trying with basis element limit: 20
2
o19 = (map(R,R,{- 3x + 9x + x , x , 3x - 2x + x , x }), ideal (- 2x +
1 2 4 1 1 2 3 2 1
-----------------------------------------------------------------------
3 2 2 3 2 2
9x x + x x + 1, - 9x x + 33x x - 18x x - 3x x x + 9x x x +
1 2 1 4 1 2 1 2 1 2 1 2 3 1 2 3
-----------------------------------------------------------------------
2 2
3x x x - 2x x x + x x x x + 1), {x , x })
1 2 4 1 2 4 1 2 3 4 4 3
o19 : Sequence
|
This symbol is provided by the package NoetherNormalization.