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
4 7 2
o3 = (map(R,R,{-x + x + x , x , 2x + 2x + x , x }), ideal (-x + x x +
3 1 2 4 1 1 2 3 2 3 1 1 2
------------------------------------------------------------------------
8 3 14 2 2 3 4 2 2 2 2
x x + 1, -x x + --x x + 2x x + -x x x + x x x + 2x x x + 2x x x
1 4 3 1 2 3 1 2 1 2 3 1 2 3 1 2 3 1 2 4 1 2 4
------------------------------------------------------------------------
+ x x x x + 1), {x , x })
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
7 3 3 7
o6 = (map(R,R,{-x + -x + x , x , x + -x + x , 3x + -x + x , x }), ideal
6 1 4 2 5 1 1 7 2 4 1 8 2 3 2
------------------------------------------------------------------------
7 2 3 3 343 3 49 2 2 49 2 63 3
(-x + -x x + x x - x , ---x x + --x x + --x x x + --x x +
6 1 4 1 2 1 5 2 216 1 2 16 1 2 12 1 2 5 32 1 2
------------------------------------------------------------------------
21 2 7 2 27 4 27 3 9 2 2 3
--x x x + -x x x + --x + --x x + -x x + x x ), {x , x , x })
4 1 2 5 2 1 2 5 64 2 16 2 5 4 2 5 2 5 5 4 3
o6 : Sequence
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i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 21504x_1x_2x_5^6-84672x_2^9x_5-5103x_2^9+56448x_2^
{-9} | 3402x_1x_2^2x_5^3-37632x_1x_2x_5^5+4536x_1x_2x_5^4
{-9} | 22320522x_1x_2^3+246903552x_1x_2^2x_5^2+59521392x_
{-3} | 14x_1^2+9x_1x_2+12x_1x_5-12x_2^3
------------------------------------------------------------------------
8x_5^2+6804x_2^8x_5-25088x_2^7x_5^3-9072x_2^7x_5^2+12096x_2^6x_5^3-
+148176x_2^9-98784x_2^8x_5-3969x_2^8+43904x_2^7x_5^2+10584x_2^7x_5-
1x_2^2x_5+15105785856x_1x_2x_5^5-910393344x_1x_2x_5^4+219469824x_1x
------------------------------------------------------------------------
16128x_2^5x_5^4+21504x_2^4x_5^5+13824x_2^2x_5^6+18432x_2x_5^7
21168x_2^6x_5^2+28224x_2^5x_5^3-37632x_2^4x_5^4+4536x_2^4x_5^3+
_2x_5^3+39680928x_1x_2x_5^2-59479031808x_2^9+39652687872x_2^8x_
------------------------------------------------------------------------
2187x_2^3x_5^3-24192x_2^2x_5^5+5832x_2^2x_5^4-32256x_2x_5^6+3888x_2x_5^5
5+2389782528x_2^8-17623416832x_2^7x_5^2-5310627840x_2^7x_5+128024064x_2^
------------------------------------------------------------------------
7+8497004544x_2^6x_5^2-512096256x_2^6x_5-61725888x_2^6-11329339392x_2^5x
------------------------------------------------------------------------
_5^3+682795008x_2^5x_5^2+82301184x_2^5x_5+29760696x_2^5+15105785856x_2^
------------------------------------------------------------------------
4x_5^4-910393344x_2^4x_5^3+219469824x_2^4x_5^2+39680928x_2^4x_5+
------------------------------------------------------------------------
14348907x_2^4+158723712x_2^3x_5^2+57395628x_2^3x_5+9710862336x_2^2x_5^5-
------------------------------------------------------------------------
585252864x_2^2x_5^4+352719360x_2^2x_5^3+76527504x_2^2x_5^2+12947816448x_
------------------------------------------------------------------------
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|
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2x_5^6-780337152x_2x_5^5+188116992x_2x_5^4+34012224x_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
7 2 7 2 7
o13 = (map(R,R,{3x + -x + x , x , -x + -x + x , x }), ideal (4x + -x x
1 9 2 4 1 9 1 5 2 3 2 1 9 1 2
-----------------------------------------------------------------------
2 3 1771 2 2 49 3 2 7 2 2 2
+ x x + 1, -x x + ----x x + --x x + 3x x x + -x x x + -x x x +
1 4 3 1 2 405 1 2 45 1 2 1 2 3 9 1 2 3 9 1 2 4
-----------------------------------------------------------------------
7 2
-x x x + x x x x + 1), {x , x })
5 1 2 4 1 2 3 4 4 3
o13 : Sequence
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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
3 2 7 5 13 2
o16 = (map(R,R,{--x + -x + x , x , -x + -x + x , x }), ideal (--x +
10 1 7 2 4 1 6 1 9 2 3 2 10 1
-----------------------------------------------------------------------
2 7 3 1 2 2 10 3 3 2 2 2
-x x + x x + 1, --x x + -x x + --x x + --x x x + -x x x +
7 1 2 1 4 20 1 2 2 1 2 63 1 2 10 1 2 3 7 1 2 3
-----------------------------------------------------------------------
7 2 5 2
-x x x + -x x x + x x x x + 1), {x , x })
6 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];
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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
2
o19 = (map(R,R,{x + 2x + x , x , 2x - x + x , x }), ideal (2x + 2x x +
1 2 4 1 1 2 3 2 1 1 2
-----------------------------------------------------------------------
3 2 2 3 2 2 2 2
x x + 1, 2x x + 3x x - 2x x + x x x + 2x x x + 2x x x - x x x +
1 4 1 2 1 2 1 2 1 2 3 1 2 3 1 2 4 1 2 4
-----------------------------------------------------------------------
x x x x + 1), {x , x })
1 2 3 4 4 3
o19 : Sequence
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This symbol is provided by the package NoetherNormalization.