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];
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i2 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);
o2 : Ideal of R
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i3 : (f,J,X) = noetherNormalization I
3 1 2 3
o3 = (map(R,R,{x + -x + x , x , 4x + -x + x , x }), ideal (2x + -x x +
1 4 2 4 1 1 2 2 3 2 1 4 1 2
------------------------------------------------------------------------
3 7 2 2 3 3 2 3 2 2 1 2
x x + 1, 4x x + -x x + -x x + x x x + -x x x + 4x x x + -x x x +
1 4 1 2 2 1 2 8 1 2 1 2 3 4 1 2 3 1 2 4 2 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];
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i5 : I = ideal(x_2*x_1-x_5^3, x_5*x_1^3);
o5 : Ideal of R
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i6 : (f,J,X) = noetherNormalization I
10 6 10 1 4
o6 = (map(R,R,{--x + -x + x , x , --x + -x + x , 9x + -x + x , x }),
9 1 7 2 5 1 3 1 5 2 4 1 5 2 3 2
------------------------------------------------------------------------
10 2 6 3 1000 3 200 2 2 100 2
ideal (--x + -x x + x x - x , ----x x + ---x x + ---x x x +
9 1 7 1 2 1 5 2 729 1 2 63 1 2 27 1 2 5
------------------------------------------------------------------------
120 3 40 2 10 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 ,
49 1 2 7 1 2 5 3 1 2 5 343 2 49 2 5 7 2 5 2 5 5
------------------------------------------------------------------------
x , x })
4 3
o6 : Sequence
|
i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 1512630x_1x_2x_5^
{-9} | 1224720x_1x_2^2x_
{-9} | 1157095860480x_1x
{-3} | 70x_1^2+54x_1x_2+
------------------------------------------------------------------------
6-7408800x_2^9x_5-699840x_2^9+4321800x_2^8x_5^2+816480x_
5^3-7563150x_1x_2x_5^5+1428840x_1x_2x_5^4+37044000x_2^9-
_2^3+7145543109600x_1x_2^2x_5^2+2699890341120x_1x_2^2x_5
63x_1x_5-63x_2^3
------------------------------------------------------------------------
2^8x_5-1680700x_2^7x_5^3-
21609000x_2^8x_5-1360800x
+155714481011250x_1x_2x_5
------------------------------------------------------------------------
952560x_2^7x_5^2+1111320x_2^6x_5^3-1296540x_2^5x_5^4+1512630x_2^4x_5^5+
_2^8+8403500x_2^7x_5^2+3175200x_2^7x_5-5556600x_2^6x_5^2+6482700x_2^5x_
^5-14708889751500x_1x_2x_5^4+5557644640800x_1x_2x_5^3+1574936032320x_1x
------------------------------------------------------------------------
1166886x_2^2x_5^6+1361367x_2x_5^7
5^3-7563150x_2^4x_5^4+1428840x_2^4x_5^3+944784x_2^3x_5^3-5834430x_2^2x_5
_2x_5^2-762683172300000x_2^9+444898517175000x_2^8x_5+42025399290000x_2^8
------------------------------------------------------------------------
^5+2204496x_2^2x_5^4-6806835x_2x_5^6+1285956x_2x_5^5
-173016090012500x_2^7x_5^2-81716054175000x_2^7x_5+3087580356000x_2^7+
------------------------------------------------------------------------
114402475845000x_2^6x_5^2-10806531246000x_2^6x_5-2041583745600x_2^6-
------------------------------------------------------------------------
133469555152500x_2^5x_5^3+12607619787000x_2^5x_5^2+2381847703200x_2^5x_5
------------------------------------------------------------------------
+1349945170560x_2^5+155714481011250x_2^4x_5^4-14708889751500x_2^4x_5^3+
------------------------------------------------------------------------
5557644640800x_2^4x_5^2+1574936032320x_2^4x_5+892616806656x_2^4+
------------------------------------------------------------------------
5512276113120x_2^3x_5^2+3124158823296x_2^3x_5+120122599637250x_2^2x_5^5-
------------------------------------------------------------------------
11346857808300x_2^2x_5^4+10718314664400x_2^2x_5^3+3644851960512x_2^2x_5^
------------------------------------------------------------------------
2+140143032910125x_2x_5^6-13238000776350x_2x_5^5+5001880176720x_2x_5^4+
------------------------------------------------------------------------
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|
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1417442429088x_2x_5^3 |
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5 1
o7 : Matrix R <--- R
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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];
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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 2 5 3 11 2 2
o13 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (--x + -x x
8 1 3 2 4 1 3 1 4 2 3 2 8 1 3 1 2
-----------------------------------------------------------------------
5 3 401 2 2 1 3 3 2 2 2 5 2
+ x x + 1, -x x + ---x x + -x x + -x x x + -x x x + -x x x +
1 4 8 1 2 288 1 2 2 1 2 8 1 2 3 3 1 2 3 3 1 2 4
-----------------------------------------------------------------------
3 2
-x x x + x x x x + 1), {x , x })
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
3 4 4 2
o16 = (map(R,R,{10x + -x + x , x , -x + -x + x , x }), ideal (11x +
1 4 2 4 1 5 1 9 2 3 2 1
-----------------------------------------------------------------------
3 3 227 2 2 1 3 2 3 2
-x x + x x + 1, 8x x + ---x x + -x x + 10x x x + -x x x +
4 1 2 1 4 1 2 45 1 2 3 1 2 1 2 3 4 1 2 3
-----------------------------------------------------------------------
4 2 4 2
-x x x + -x x x + x x x x + 1), {x , x })
5 1 2 4 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 + 2x + x , x }), ideal (x + 2x x + x x
2 4 1 1 2 3 2 1 1 2 1 4
-----------------------------------------------------------------------
2 2 3 2 2 2
+ 1, - 2x x + 4x x + 2x x x - x x x + 2x x x + x x x x + 1), {x ,
1 2 1 2 1 2 3 1 2 4 1 2 4 1 2 3 4 4
-----------------------------------------------------------------------
x })
3
o19 : Sequence
|
This symbol is provided by the package NoetherNormalization.