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 9 7 2 5 2 9
o3 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (-x + -x x +
2 1 7 2 4 1 6 1 5 2 3 2 2 1 7 1 2
------------------------------------------------------------------------
7 3 21 2 2 18 3 3 2 9 2 7 2
x x + 1, -x x + --x x + --x x + -x x x + -x x x + -x x x +
1 4 4 1 2 10 1 2 35 1 2 2 1 2 3 7 1 2 3 6 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
o3 : Sequence
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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
2 6 5 4 5
o6 = (map(R,R,{-x + -x + x , x , -x + x + x , -x + -x + x , x }), ideal
9 1 7 2 5 1 7 1 2 4 9 1 2 2 3 2
------------------------------------------------------------------------
2 2 6 3 8 3 8 2 2 4 2 24 3 8 2
(-x + -x x + 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 63 1 2 27 1 2 5 49 1 2 7 1 2 5
------------------------------------------------------------------------
2 2 216 4 108 3 18 2 2 3
+ -x x x + ---x + ---x x + --x x + x x ), {x , x , x })
3 1 2 5 343 2 49 2 5 7 2 5 2 5 5 4 3
o6 : Sequence
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i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 302526x_1x_2x_5^6-296352x_2^9x_5-139968x_2^9+
{-9} | 244944x_1x_2^2x_5^3-302526x_1x_2x_5^5+285768x
{-9} | 231419172096x_1x_2^3+285821724384x_1x_2^2x_5^
{-3} | 14x_1^2+54x_1x_2+63x_1x_5-63x_2^3
------------------------------------------------------------------------
172872x_2^8x_5^2+163296x_2^8x_5-67228x_2^7x_5^3-190512x_2^7x_5^2+222264x
_1x_2x_5^4+296352x_2^9-172872x_2^8x_5-54432x_2^8+67228x_2^7x_5^2+127008x
2+539978068224x_1x_2^2x_5+249143169618x_1x_2x_5^5-117671118012x_1x_2x_5^
------------------------------------------------------------------------
_2^6x_5^3-259308x_2^5x_5^4+302526x_2^4x_5^
_2^7x_5-222264x_2^6x_5^2+259308x_2^5x_5^3-
4+222305785632x_1x_2x_5^3+314987206464x_1x
------------------------------------------------------------------------
5+1166886x_2^2x_5^6+1361367x_2x_5^7
302526x_2^4x_5^4+285768x_2^4x_5^3+944784x_2^3x_5^3-1166886x_2^2x
_2x_5^2-244058615136x_2^9+142367525496x_2^8x_5+67240638864x_2^8-
------------------------------------------------------------------------
_5^5+2204496x_2^2x_5^4-1361367x_2x_5^6+1285956x_2x_5^5
55365148804x_2^7x_5^2-130745686680x_2^7x_5+24700642848x_2^7+
------------------------------------------------------------------------
183043961352x_2^6x_5^2-86452249968x_2^6x_5-81663349824x_2^6-
------------------------------------------------------------------------
213551288244x_2^5x_5^3+100860958296x_2^5x_5^2+95273908128x_2^5x_5+
------------------------------------------------------------------------
269989034112x_2^5+249143169618x_2^4x_5^4-117671118012x_2^4x_5^3+
------------------------------------------------------------------------
222305785632x_2^4x_5^2+314987206464x_2^4x_5+892616806656x_2^4+
------------------------------------------------------------------------
1102455222624x_2^3x_5^2+3124158823296x_2^3x_5+960980797098x_2^2x_5^5-
------------------------------------------------------------------------
453874312332x_2^2x_5^4+2143662932880x_2^2x_5^3+3644851960512x_2^2x_5^2+
------------------------------------------------------------------------
1121144263281x_2x_5^6-529520031054x_2x_5^5+1000376035344x_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
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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
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Here is an example with the option
Verbose => true:
i11 : R = QQ[x_1..x_4];
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i12 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);
o12 : Ideal of R
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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 1 13 2
o13 = (map(R,R,{--x + 7x + x , x , x + -x + x , x }), ideal (--x + 7x x
3 1 2 4 1 1 2 2 3 2 3 1 1 2
-----------------------------------------------------------------------
10 3 26 2 2 7 3 10 2 2 2
+ x x + 1, --x x + --x x + -x x + --x x x + 7x x x + x x x +
1 4 3 1 2 3 1 2 2 1 2 3 1 2 3 1 2 3 1 2 4
-----------------------------------------------------------------------
1 2
-x x x + x x x x + 1), {x , x })
2 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];
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i15 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);
o15 : Ideal of R
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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
8 5 11 2
o16 = (map(R,R,{-x + 5x + x , x , 5x + -x + x , x }), ideal (--x + 5x x
3 1 2 4 1 1 8 2 3 2 3 1 1 2
-----------------------------------------------------------------------
40 3 80 2 2 25 3 8 2 2 2
+ x x + 1, --x x + --x x + --x x + -x x x + 5x x x + 5x x x +
1 4 3 1 2 3 1 2 8 1 2 3 1 2 3 1 2 3 1 2 4
-----------------------------------------------------------------------
5 2
-x x x + x x x x + 1), {x , x })
8 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];
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i18 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);
o18 : Ideal of R
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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 - 4x + x , x , 2x + 2x + x , x }), ideal (- 3x -
1 2 4 1 1 2 3 2 1
-----------------------------------------------------------------------
3 2 2 3 2 2
4x x + x x + 1, - 8x x - 16x x - 8x x - 4x x x - 4x x x +
1 2 1 4 1 2 1 2 1 2 1 2 3 1 2 3
-----------------------------------------------------------------------
2 2
2x 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
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This symbol is provided by the package NoetherNormalization.