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 3 2 11 2 3
o3 = (map(R,R,{-x + -x + x , x , -x + 2x + x , x }), ideal (--x + -x x
9 1 2 2 4 1 3 1 2 3 2 9 1 2 1 2
------------------------------------------------------------------------
4 3 13 2 2 3 2 2 3 2 2 2
+ x x + 1, --x x + --x x + 3x x + -x x x + -x x x + -x x x +
1 4 27 1 2 9 1 2 1 2 9 1 2 3 2 1 2 3 3 1 2 4
------------------------------------------------------------------------
2
2x 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
7 1 7 7
o6 = (map(R,R,{-x + -x + x , x , --x + 7x + x , 3x + -x + x , x }),
8 1 5 2 5 1 10 1 2 4 1 8 2 3 2
------------------------------------------------------------------------
7 2 1 3 343 3 147 2 2 147 2 21 3
ideal (-x + -x x + x x - x , ---x x + ---x x + ---x x x + ---x x
8 1 5 1 2 1 5 2 512 1 2 320 1 2 64 1 2 5 200 1 2
------------------------------------------------------------------------
21 2 21 2 1 4 3 3 3 2 2 3
+ --x x x + --x x x + ---x + --x x + -x x + x x ), {x , x , x })
20 1 2 5 8 1 2 5 125 2 25 2 5 5 2 5 2 5 5 4 3
o6 : Sequence
|
i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 175000x_1x_2x_5^6-36750x_2^9x_5-56x_2^9+91875x_2^8x_5^
{-9} | 2240x_1x_2^2x_5^3-3675000x_1x_2x_5^5+11200x_1x_2x_5^4+
{-9} | 71680x_1x_2^3+117600000x_1x_2^2x_5^2+716800x_1x_2^2x_5
{-3} | 35x_1^2+8x_1x_2+40x_1x_5-40x_2^3
------------------------------------------------------------------------
2+280x_2^8x_5-153125x_2^7x_5^3-1400x_2^7x_5^2+7000x_2^6x_5^3-35000x_2^
771750x_2^9-1929375x_2^8x_5-1960x_2^8+3215625x_2^7x_5^2+19600x_2^7x_5-
+42205078125000x_1x_2x_5^5-64312500000x_1x_2x_5^4+392000000x_1x_2x_5^3
------------------------------------------------------------------------
5x_5^4+175000x_2^4x_5^5+40000x_2^2x_5^6+200000x_2x_5^7
147000x_2^6x_5^2+735000x_2^5x_5^3-3675000x_2^4x_5^4+11200x_2^
+1792000x_1x_2x_5^2-8863066406250x_2^9+22157666015625x_2^8x_5
------------------------------------------------------------------------
4x_5^3+512x_2^3x_5^3-840000x_2^2x_5^5+5120x_2^2x_5^4-4200000x_2x
+33764062500x_2^8-36929443359375x_2^7x_5^2-281367187500x_2^7x_5+
------------------------------------------------------------------------
_5^6+12800x_2x_5^5
171500000x_2^7+1688203125000x_2^6x_5^2-2572500000x_2^6x_5-7840000x_2^6-
------------------------------------------------------------------------
8441015625000x_2^5x_5^3+12862500000x_2^5x_5^2+39200000x_2^5x_5+358400x_2
------------------------------------------------------------------------
^5+42205078125000x_2^4x_5^4-64312500000x_2^4x_5^3+392000000x_2^4x_5^2+
------------------------------------------------------------------------
1792000x_2^4x_5+16384x_2^4+26880000x_2^3x_5^2+245760x_2^3x_5+
------------------------------------------------------------------------
9646875000000x_2^2x_5^5-14700000000x_2^2x_5^4+224000000x_2^2x_5^3+
------------------------------------------------------------------------
1228800x_2^2x_5^2+48234375000000x_2x_5^6-73500000000x_2x_5^5+448000000x_
------------------------------------------------------------------------
|
|
|
2x_5^4+2048000x_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 7 10 11 2
o13 = (map(R,R,{-x + -x + x , x , 8x + --x + x , x }), ideal (--x +
8 1 5 2 4 1 1 9 2 3 2 8 1
-----------------------------------------------------------------------
7 3 697 2 2 14 3 3 2 7 2
-x x + x x + 1, 3x x + ---x x + --x x + -x x x + -x x x +
5 1 2 1 4 1 2 60 1 2 9 1 2 8 1 2 3 5 1 2 3
-----------------------------------------------------------------------
2 10 2
8x x x + --x x x + x x x x + 1), {x , x })
1 2 4 9 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 9 7 13 2
o16 = (map(R,R,{--x + -x + x , x , -x + --x + x , x }), ideal (--x +
10 1 5 2 4 1 5 1 10 2 3 2 10 1
-----------------------------------------------------------------------
1 27 3 57 2 2 7 3 3 2 1 2
-x x + x x + 1, --x x + ---x x + --x x + --x x x + -x x x +
5 1 2 1 4 50 1 2 100 1 2 50 1 2 10 1 2 3 5 1 2 3
-----------------------------------------------------------------------
9 2 7 2
-x x x + --x x x + x x x x + 1), {x , x })
5 1 2 4 10 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,{4x - x + x , x , 4x - 5x + x , x }), ideal (5x - x x +
1 2 4 1 1 2 3 2 1 1 2
-----------------------------------------------------------------------
3 2 2 3 2 2 2
x x + 1, 16x x - 24x x + 5x x + 4x x x - x x x + 4x x x -
1 4 1 2 1 2 1 2 1 2 3 1 2 3 1 2 4
-----------------------------------------------------------------------
2
5x x x + x x x x + 1), {x , x })
1 2 4 1 2 3 4 4 3
o19 : Sequence
|
This symbol is provided by the package NoetherNormalization.