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
8 10 5 15 2
o3 = (map(R,R,{-x + x + x , x , --x + -x + x , x }), ideal (--x + x x +
7 1 2 4 1 9 1 9 2 3 2 7 1 1 2
------------------------------------------------------------------------
80 3 110 2 2 5 3 8 2 2 10 2
x x + 1, --x x + ---x x + -x x + -x x x + x x x + --x x x +
1 4 63 1 2 63 1 2 9 1 2 7 1 2 3 1 2 3 9 1 2 4
------------------------------------------------------------------------
5 2
-x x x + x x x x + 1), {x , x })
9 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
8 2 4
o6 = (map(R,R,{-x + -x + x , x , -x + 2x + x , x + x + x , x }), ideal
9 1 5 2 5 1 5 1 2 4 1 2 3 2
------------------------------------------------------------------------
8 2 2 3 512 3 128 2 2 64 2 32 3
(-x + -x x + x x - x , ---x x + ---x x + --x x x + --x x +
9 1 5 1 2 1 5 2 729 1 2 135 1 2 27 1 2 5 75 1 2
------------------------------------------------------------------------
32 2 8 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 })
15 1 2 5 3 1 2 5 125 2 25 2 5 5 2 5 2 5 5 4 3
o6 : Sequence
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i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 225000x_1x_2x_5^6-192000x_2^9x_5-2304x_2^9+240000x_2
{-9} | 2160x_1x_2^2x_5^3-225000x_1x_2x_5^5+5400x_1x_2x_5^4+
{-9} | 116640x_1x_2^3+12150000x_1x_2^2x_5^2+583200x_1x_2^2x
{-3} | 40x_1^2+18x_1x_2+45x_1x_5-45x_2^3
------------------------------------------------------------------------
^8x_5^2+5760x_2^8x_5-200000x_2^7x_5^3-14400x_2^7x_5^2+36000x_2^6x_5
192000x_2^9-240000x_2^8x_5-1920x_2^8+200000x_2^7x_5^2+9600x_2^7x_5-
_5+35156250000x_1x_2x_5^5-421875000x_1x_2x_5^4+20250000x_1x_2x_5^3+
------------------------------------------------------------------------
^3-90000x_2^5x_5^4+225000x_2^4x_5^5+101250x_2^2x_5^6+253125x_2x_5^7
36000x_2^6x_5^2+90000x_2^5x_5^3-225000x_2^4x_5^4+5400x_2^4x_5^3+972x_2
729000x_1x_2x_5^2-30000000000x_2^9+37500000000x_2^8x_5+450000000x_2^8-
------------------------------------------------------------------------
^3x_5^3-101250x_2^2x_5^5+4860x_2^2x_5^4-253125x_2x_5^6+6075x_2x_5^5
31250000000x_2^7x_5^2-1875000000x_2^7x_5+9000000x_2^7+5625000000x_2^6x_5
------------------------------------------------------------------------
^2-67500000x_2^6x_5-1620000x_2^6-14062500000x_2^5x_5^3+168750000x_2^5x_5
------------------------------------------------------------------------
^2+4050000x_2^5x_5+291600x_2^5+35156250000x_2^4x_5^4-421875000x_2^4x_5^3
------------------------------------------------------------------------
+20250000x_2^4x_5^2+729000x_2^4x_5+52488x_2^4+5467500x_2^3x_5^2+393660x_
------------------------------------------------------------------------
2^3x_5+15820312500x_2^2x_5^5-189843750x_2^2x_5^4+22781250x_2^2x_5^3+
------------------------------------------------------------------------
984150x_2^2x_5^2+39550781250x_2x_5^6-474609375x_2x_5^5+22781250x_2x_5^4+
------------------------------------------------------------------------
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820125x_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
8 3 7 4 15 2 3
o13 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (--x + -x x
7 1 7 2 4 1 8 1 3 2 3 2 7 1 7 1 2
-----------------------------------------------------------------------
3 319 2 2 4 3 8 2 3 2 7 2
+ x x + 1, x x + ---x x + -x x + -x x x + -x x x + -x x x +
1 4 1 2 168 1 2 7 1 2 7 1 2 3 7 1 2 3 8 1 2 4
-----------------------------------------------------------------------
4 2
-x x x + x x x x + 1), {x , x })
3 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
2 2 7 2 2
o16 = (map(R,R,{3x + -x + x , x , -x + -x + x , x }), ideal (4x + -x x
1 7 2 4 1 7 1 9 2 3 2 1 7 1 2
-----------------------------------------------------------------------
6 3 355 2 2 2 3 2 2 2 2 2
+ x x + 1, -x x + ---x x + -x x + 3x x x + -x x x + -x x x +
1 4 7 1 2 147 1 2 9 1 2 1 2 3 7 1 2 3 7 1 2 4
-----------------------------------------------------------------------
7 2
-x x x + x x x x + 1), {x , x })
9 1 2 4 1 2 3 4 4 3
o16 : Sequence
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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 + x + x , x , - 2x - 2x + x , x }), ideal (2x + x 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 - 4x x - 2x x + x x x + x x x - 2x x x - 2x 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.