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
3 5 3 1 5 2 5
o3 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (-x + -x x +
2 1 3 2 4 1 5 1 8 2 3 2 2 1 3 1 2
------------------------------------------------------------------------
9 3 19 2 2 5 3 3 2 5 2 3 2
x x + 1, --x x + --x x + --x x + -x x x + -x x x + -x x x +
1 4 10 1 2 16 1 2 24 1 2 2 1 2 3 3 1 2 3 5 1 2 4
------------------------------------------------------------------------
1 2
-x x x + x x x x + 1), {x , x })
8 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
5 2 5 5 4 4
o6 = (map(R,R,{-x + -x + x , x , -x + -x + x , -x + -x + x , x }),
9 1 7 2 5 1 7 1 9 2 4 5 1 3 2 3 2
------------------------------------------------------------------------
5 2 2 3 125 3 50 2 2 25 2 20 3
ideal (-x + -x x + x x - x , ---x x + ---x x + --x x x + ---x x +
9 1 7 1 2 1 5 2 729 1 2 189 1 2 27 1 2 5 147 1 2
------------------------------------------------------------------------
20 2 5 2 8 4 12 3 6 2 2 3
--x x x + -x x x + ---x + --x x + -x x + x x ), {x , x , x })
21 1 2 5 3 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} | 756315x_1x_2x_5^6-205800x_2^9x_5-1440x_2^9+360150x_2^8x
{-9} | 15120x_1x_2^2x_5^3-3781575x_1x_2x_5^5+52920x_1x_2x_5^4+
{-9} | 26127360x_1x_2^3+6534561600x_1x_2^2x_5^2+182891520x_1x_
{-3} | 35x_1^2+18x_1x_2+63x_1x_5-63x_2^3
------------------------------------------------------------------------
_5^2+5040x_2^8x_5-420175x_2^7x_5^3-17640x_2^7x_5^2+61740x
1029000x_2^9-1800750x_2^8x_5-8400x_2^8+2100875x_2^7x_5^2+
2^2x_5+77857240505625x_1x_2x_5^5-544773694500x_1x_2x_5^4+
------------------------------------------------------------------------
_2^6x_5^3-216090x_2^5x_5^4+756315x_2^4x_5^5+388962x_2^2x_5^6
58800x_2^7x_5-308700x_2^6x_5^2+1080450x_2^5x_5^3-3781575x_2^
15247310400x_1x_2x_5^3+320060160x_1x_2x_5^2-21185643675000x_
------------------------------------------------------------------------
+1361367x_2x_5^7
4x_5^4+52920x_2^4x_5^3+7776x_2^3x_5^3-1944810x_2^2x_5^5+54432x_2^2x_5^
2^9+37074876431250x_2^8x_5+259416045000x_2^8-43254022503125x_2^7x_5^2-
------------------------------------------------------------------------
4-6806835x_2x_5^6+95256x_2x_5^5
1513260262500x_2^7x_5+4235364000x_2^7+6355693102500x_2^6x_5^2-
------------------------------------------------------------------------
44471322000x_2^6x_5-622339200x_2^6-22244925858750x_2^5x_5^3+
------------------------------------------------------------------------
155649627000x_2^5x_5^2+2178187200x_2^5x_5+91445760x_2^5+77857240505625x_
------------------------------------------------------------------------
2^4x_5^4-544773694500x_2^4x_5^3+15247310400x_2^4x_5^2+320060160x_2^4x_5+
------------------------------------------------------------------------
13436928x_2^4+3360631680x_2^3x_5^2+141087744x_2^3x_5+40040866545750x_2^
------------------------------------------------------------------------
2x_5^5-280169328600x_2^2x_5^4+19603684800x_2^2x_5^3+493807104x_2^2x_5^2+
------------------------------------------------------------------------
140143032910125x_2x_5^6-980592650100x_2x_5^5+27445158720x_2x_5^4+
------------------------------------------------------------------------
|
|
|
576108288x_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
10 9 10 17 2
o13 = (map(R,R,{--x + -x + x , x , --x + 4x + x , x }), ideal (--x +
7 1 2 2 4 1 7 1 2 3 2 7 1
-----------------------------------------------------------------------
9 100 3 85 2 2 3 10 2 9 2
-x x + x x + 1, ---x x + --x x + 18x x + --x x x + -x x x +
2 1 2 1 4 49 1 2 7 1 2 1 2 7 1 2 3 2 1 2 3
-----------------------------------------------------------------------
10 2 2
--x x x + 4x x x + x x x x + 1), {x , x })
7 1 2 4 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
2 6 1 5 2 6
o16 = (map(R,R,{-x + -x + x , x , x + -x + x , x }), ideal (-x + -x x +
3 1 5 2 4 1 1 9 2 3 2 3 1 5 1 2
-----------------------------------------------------------------------
2 3 172 2 2 2 3 2 2 6 2 2
x x + 1, -x x + ---x x + --x x + -x x x + -x x x + x x x +
1 4 3 1 2 135 1 2 15 1 2 3 1 2 3 5 1 2 3 1 2 4
-----------------------------------------------------------------------
1 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
--trying with basis element limit: 20
2
o19 = (map(R,R,{- 3x - 2x + x , x , 4x - 4x + x , x }), ideal (- 2x -
1 2 4 1 1 2 3 2 1
-----------------------------------------------------------------------
3 2 2 3 2 2
2x x + x x + 1, - 12x x + 4x x + 8x x - 3x x x - 2x x x +
1 2 1 4 1 2 1 2 1 2 1 2 3 1 2 3
-----------------------------------------------------------------------
2 2
4x x x - 4x 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.