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 8 2 3
o3 = (map(R,R,{2x + -x + x , x , --x + -x + x , x }), ideal (3x + -x x
1 5 2 4 1 10 1 5 2 3 2 1 5 1 2
------------------------------------------------------------------------
9 3 187 2 2 24 3 2 3 2 9 2
+ x x + 1, -x x + ---x x + --x x + 2x x x + -x x x + --x x x +
1 4 5 1 2 50 1 2 25 1 2 1 2 3 5 1 2 3 10 1 2 4
------------------------------------------------------------------------
8 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
1 1 8 4 2
o6 = (map(R,R,{-x + -x + x , x , -x + x + x , -x + -x + x , x }), ideal
3 1 3 2 5 1 7 1 2 4 9 1 3 2 3 2
------------------------------------------------------------------------
1 2 1 3 1 3 1 2 2 1 2 1 3 2 2
(-x + -x x + x x - x , --x x + -x x + -x x x + -x x + -x x x +
3 1 3 1 2 1 5 2 27 1 2 9 1 2 3 1 2 5 9 1 2 3 1 2 5
------------------------------------------------------------------------
2 1 4 1 3 2 2 3
x x x + --x + -x x + x x + x x ), {x , x , x })
1 2 5 27 2 3 2 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} | 243x_1x_2x_5^6-54x_2^9x_5-x_2^9+81x_2^8x_5^2+3x_2^8x_5-81x_2^7x_
{-9} | 3x_1x_2^2x_5^3-243x_1x_2x_5^5+9x_1x_2x_5^4+54x_2^9-81x_2^8x_5-x_
{-9} | x_1x_2^3+81x_1x_2^2x_5^2+6x_1x_2^2x_5+118098x_1x_2x_5^5-2187x_1x
{-3} | x_1^2+x_1x_2+3x_1x_5-3x_2^3
------------------------------------------------------------------------
5^3-9x_2^7x_5^2+27x_2^6x_5^3-81x_2^5x_5^4+243x_2^4x_5^5+243x_2^2x_5^6+
2^8+81x_2^7x_5^2+6x_2^7x_5-27x_2^6x_5^2+81x_2^5x_5^3-243x_2^4x_5^4+9x_
_2x_5^4+162x_1x_2x_5^3+9x_1x_2x_5^2-26244x_2^9+39366x_2^8x_5+729x_2^8-
------------------------------------------------------------------------
729x_2x_5^7
2^4x_5^3+3x_2^3x_5^3-243x_2^2x_5^5+18x_2^2x_5^4-729x_2x_5^6+27x_2x_5^5
39366x_2^7x_5^2-3645x_2^7x_5+27x_2^7+13122x_2^6x_5^2-243x_2^6x_5-9x_2^6-
------------------------------------------------------------------------
39366x_2^5x_5^3+729x_2^5x_5^2+27x_2^5x_5+3x_2^5+118098x_2^4x_5^4-2187x_2
------------------------------------------------------------------------
^4x_5^3+162x_2^4x_5^2+9x_2^4x_5+x_2^4+81x_2^3x_5^2+9x_2^3x_5+118098x_2^
------------------------------------------------------------------------
2x_5^5-2187x_2^2x_5^4+405x_2^2x_5^3+27x_2^2x_5^2+354294x_2x_5^6-6561x_2x
------------------------------------------------------------------------
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_5^5+486x_2x_5^4+27x_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
9 2 9
o13 = (map(R,R,{x + -x + x , x , 3x + 10x + x , x }), ideal (2x + -x x
1 2 2 4 1 1 2 3 2 1 2 1 2
-----------------------------------------------------------------------
3 47 2 2 3 2 9 2 2
+ x x + 1, 3x x + --x x + 45x x + x x x + -x x x + 3x x x +
1 4 1 2 2 1 2 1 2 1 2 3 2 1 2 3 1 2 4
-----------------------------------------------------------------------
2
10x x x + x x x x + 1), {x , x })
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
7 4 9 8 11 2 4
o16 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (--x + -x x
4 1 5 2 4 1 5 1 9 2 3 2 4 1 5 1 2
-----------------------------------------------------------------------
63 3 674 2 2 32 3 7 2 4 2 9 2
+ x x + 1, --x x + ---x x + --x x + -x x x + -x x x + -x x x +
1 4 20 1 2 225 1 2 45 1 2 4 1 2 3 5 1 2 3 5 1 2 4
-----------------------------------------------------------------------
8 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
--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,{2x - x + x , x , - 4x - 5x + x , x }), ideal (3x - x x
1 2 4 1 1 2 3 2 1 1 2
-----------------------------------------------------------------------
3 2 2 3 2 2 2
+ x x + 1, - 8x x - 6x x + 5x x + 2x x x - x x x - 4x x x -
1 4 1 2 1 2 1 2 1 2 3 1 2 3 1 2 4
-----------------------------------------------------------------------
2
5x x x + x x x x + 1), {x , x })
1 2 4 1 2 3 4 4 3
o19 : Sequence
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This symbol is provided by the package NoetherNormalization.