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 5 5 17 2 5
o3 = (map(R,R,{-x + -x + x , x , -x + x + x , x }), ideal (--x + -x x +
9 1 2 2 4 1 2 1 2 3 2 9 1 2 1 2
------------------------------------------------------------------------
20 3 257 2 2 5 3 8 2 5 2 5 2
x x + 1, --x x + ---x x + -x x + -x x x + -x x x + -x x x +
1 4 9 1 2 36 1 2 2 1 2 9 1 2 3 2 1 2 3 2 1 2 4
------------------------------------------------------------------------
2
x x x + x x x x + 1), {x , x })
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 8 2 10 1
o6 = (map(R,R,{-x + x + x , x , -x + -x + x , --x + -x + x , x }),
2 1 2 5 1 3 1 7 2 4 9 1 2 2 3 2
------------------------------------------------------------------------
1 2 3 1 3 3 2 2 3 2 3 3 2
ideal (-x + x x + x x - x , -x x + -x x + -x x x + -x x + 3x x x
2 1 1 2 1 5 2 8 1 2 4 1 2 4 1 2 5 2 1 2 1 2 5
------------------------------------------------------------------------
3 2 4 3 2 2 3
+ -x x x + x + 3x x + 3x x + x x ), {x , x , x })
2 1 2 5 2 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} | 2x_1x_2x_5^6-6x_2^9x_5-2x_2^9+3x_2^8x_5^2+2x_2^8x_5-x_2^7x_5^3-
{-9} | 4x_1x_2^2x_5^3-6x_1x_2x_5^5+4x_1x_2x_5^4+18x_2^9-9x_2^8x_5-2x_2
{-9} | 8x_1x_2^3+12x_1x_2^2x_5^2+16x_1x_2^2x_5+18x_1x_2x_5^5-6x_1x_2x_
{-3} | x_1^2+2x_1x_2+2x_1x_5-2x_2^3
------------------------------------------------------------------------
2x_2^7x_5^2+2x_2^6x_5^3-2x_2^5x_5^4+2x_2^4x_5^5+4x_2^2x_5^6+4x_2x_5^7
^8+3x_2^7x_5^2+4x_2^7x_5-6x_2^6x_5^2+6x_2^5x_5^3-6x_2^4x_5^4+4x_2^4x_5^3
5^4+8x_1x_2x_5^3+8x_1x_2x_5^2-54x_2^9+27x_2^8x_5+9x_2^8-9x_2^7x_5^2-15x_
------------------------------------------------------------------------
+8x_2^3x_5^3-12x_2^2x_5^5+16x_2^2x_5^4-12x_2x_5^6+8x_2x_5^5
2^7x_5+2x_2^7+18x_2^6x_5^2-6x_2^6x_5-4x_2^6-18x_2^5x_5^3+6x_2^5x_5^2+4x_
------------------------------------------------------------------------
2^5x_5+8x_2^5+18x_2^4x_5^4-6x_2^4x_5^3+8x_2^4x_5^2+8x_2^4x_5+16x_2^4+24x
------------------------------------------------------------------------
_2^3x_5^2+48x_2^3x_5+36x_2^2x_5^5-12x_2^2x_5^4+40x_2^2x_5^3+48x_2^2x_5^2
------------------------------------------------------------------------
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+36x_2x_5^6-12x_2x_5^5+16x_2x_5^4+16x_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,{a + 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
1 11 2
o13 = (map(R,R,{--x + 5x + x , x , 7x + 2x + x , x }), ideal (--x +
10 1 2 4 1 1 2 3 2 10 1
-----------------------------------------------------------------------
7 3 176 2 2 3 1 2 2
5x x + x x + 1, --x x + ---x x + 10x x + --x x x + 5x x x +
1 2 1 4 10 1 2 5 1 2 1 2 10 1 2 3 1 2 3
-----------------------------------------------------------------------
2 2
7x x x + 2x x x + x x x x + 1), {x , x })
1 2 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
2 5 7 5 2 5
o16 = (map(R,R,{-x + -x + x , x , -x + 5x + x , x }), ideal (-x + -x x
3 1 4 2 4 1 5 1 2 3 2 3 1 4 1 2
-----------------------------------------------------------------------
14 3 61 2 2 25 3 2 2 5 2 7 2
+ x x + 1, --x x + --x x + --x x + -x x x + -x x x + -x x x +
1 4 15 1 2 12 1 2 4 1 2 3 1 2 3 4 1 2 3 5 1 2 4
-----------------------------------------------------------------------
2
5x x x + x x x x + 1), {x , x })
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
2
o19 = (map(R,R,{x + x , x , 6x + x , x }), ideal (x + x x + x x + 1,
2 4 1 2 3 2 1 1 2 1 4
-----------------------------------------------------------------------
3 2 2
6x x + x x x + 6x x x + x x x x + 1), {x , x })
1 2 1 2 3 1 2 4 1 2 3 4 4 3
o19 : Sequence
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This symbol is provided by the package NoetherNormalization.