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 7 3 2 17 2 7
o3 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (--x + -x x
9 1 8 2 4 1 2 1 3 2 3 2 9 1 8 1 2
------------------------------------------------------------------------
4 3 823 2 2 7 3 8 2 7 2 3 2
+ x x + 1, -x x + ---x x + --x x + -x x x + -x x x + -x x x +
1 4 3 1 2 432 1 2 12 1 2 9 1 2 3 8 1 2 3 2 1 2 4
------------------------------------------------------------------------
2 2
-x x x + x x x x + 1), {x , x })
3 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
5 1 1 8
o6 = (map(R,R,{x + -x + x , x , -x + x + x , -x + -x + x , x }), ideal
1 2 2 5 1 7 1 2 4 4 1 5 2 3 2
------------------------------------------------------------------------
2 5 3 3 15 2 2 2 75 3 2
(x + -x x + x x - x , x x + --x x + 3x x x + --x x + 15x x x +
1 2 1 2 1 5 2 1 2 2 1 2 1 2 5 4 1 2 1 2 5
------------------------------------------------------------------------
2 125 4 75 3 15 2 2 3
3x x x + ---x + --x x + --x x + x x ), {x , x , x })
1 2 5 8 2 4 2 5 2 2 5 2 5 5 4 3
o6 : Sequence
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i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 32x_1x_2x_5^6-1200x_2^9x_5-3125x_2^9+240x_2^8x_5^2+1250x_2^8x_5-
{-9} | 1250x_1x_2^2x_5^3-96x_1x_2x_5^5+500x_1x_2x_5^4+3600x_2^9-720x_2^
{-9} | 97656250x_1x_2^3+7500000x_1x_2^2x_5^2+78125000x_1x_2^2x_5+73728x
{-3} | 2x_1^2+5x_1x_2+2x_1x_5-2x_2^3
------------------------------------------------------------------------
32x_2^7x_5^3-500x_2^7x_5^2+200x_2^6x_5^3-80x_2^5x_5^4+32x_2^4x_5^5+
8x_5-1250x_2^8+96x_2^7x_5^2+1000x_2^7x_5-600x_2^6x_5^2+240x_2^5x_5^
_1x_2x_5^5-192000x_1x_2x_5^4+2000000x_1x_2x_5^3+15625000x_1x_2x_5^2
------------------------------------------------------------------------
80x_2^2x_5^6+32x_2x_5^7
3-96x_2^4x_5^4+500x_2^4x_5^3+3125x_2^3x_5^3-240x_2^2x_5^5+2500x_2^2x_5^4
-2764800x_2^9+552960x_2^8x_5+1440000x_2^8-73728x_2^7x_5^2-960000x_2^7x_5
------------------------------------------------------------------------
-96x_2x_5^6+500x_2x_5^5
+1000000x_2^7+460800x_2^6x_5^2-1200000x_2^6x_5-6250000x_2^6-184320x_2^5x
------------------------------------------------------------------------
_5^3+480000x_2^5x_5^2+2500000x_2^5x_5+39062500x_2^5+73728x_2^4x_5^4-
------------------------------------------------------------------------
192000x_2^4x_5^3+2000000x_2^4x_5^2+15625000x_2^4x_5+244140625x_2^4+
------------------------------------------------------------------------
18750000x_2^3x_5^2+292968750x_2^3x_5+184320x_2^2x_5^5-480000x_2^2x_5^4+
------------------------------------------------------------------------
12500000x_2^2x_5^3+117187500x_2^2x_5^2+73728x_2x_5^6-192000x_2x_5^5+
------------------------------------------------------------------------
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2000000x_2x_5^4+15625000x_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
5 1 7 7 2 1
o13 = (map(R,R,{-x + -x + x , x , 3x + -x + x , x }), ideal (-x + -x x
2 1 3 2 4 1 1 4 2 3 2 2 1 3 1 2
-----------------------------------------------------------------------
15 3 43 2 2 7 3 5 2 1 2 2
+ x x + 1, --x x + --x x + --x x + -x x x + -x x x + 3x x x +
1 4 2 1 2 8 1 2 12 1 2 2 1 2 3 3 1 2 3 1 2 4
-----------------------------------------------------------------------
7 2
-x x x + x x x x + 1), {x , x })
4 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
1 1 4 2 1
o16 = (map(R,R,{2x + -x + x , x , -x + -x + x , x }), ideal (3x + -x x
1 2 2 4 1 3 1 5 2 3 2 1 2 1 2
-----------------------------------------------------------------------
2 3 53 2 2 2 3 2 1 2 1 2
+ x x + 1, -x x + --x x + -x x + 2x x x + -x x x + -x x x +
1 4 3 1 2 30 1 2 5 1 2 1 2 3 2 1 2 3 3 1 2 4
-----------------------------------------------------------------------
4 2
-x x x + x x x x + 1), {x , x })
5 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
--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 + x + x , x , - 2x - x + x , x }), ideal (5x + x x +
1 2 4 1 1 2 3 2 1 1 2
-----------------------------------------------------------------------
3 2 2 3 2 2 2 2
x x + 1, - 8x x - 6x x - x x + 4x x x + x 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.