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
10 7 1 13 2 7
o3 = (map(R,R,{--x + -x + x , x , -x + 5x + x , x }), ideal (--x + -x x
3 1 3 2 4 1 2 1 2 3 2 3 1 3 1 2
------------------------------------------------------------------------
5 3 107 2 2 35 3 10 2 7 2 1 2
+ x x + 1, -x x + ---x x + --x x + --x x x + -x x x + -x x x +
1 4 3 1 2 6 1 2 3 1 2 3 1 2 3 3 1 2 3 2 1 2 4
------------------------------------------------------------------------
2
5x 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
7 1 1 4
o6 = (map(R,R,{-x + x + x , x , 5x + -x + x , -x + -x + x , x }), ideal
3 1 2 5 1 1 4 2 4 3 1 7 2 3 2
------------------------------------------------------------------------
7 2 3 343 3 49 2 2 49 2 3 2
(-x + x x + x x - x , ---x x + --x x + --x x x + 7x x + 14x x x
3 1 1 2 1 5 2 27 1 2 3 1 2 3 1 2 5 1 2 1 2 5
------------------------------------------------------------------------
2 4 3 2 2 3
+ 7x x x + x + 3x x + 3x x + x x ), {x , x , x })
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} | 21x_1x_2x_5^6-294x_2^9x_5-21x_2^9+147x_2^8x_5^2+21x_2
{-9} | 21x_1x_2^2x_5^3-147x_1x_2x_5^5+21x_1x_2x_5^4+2058x_2^
{-9} | 63x_1x_2^3+441x_1x_2^2x_5^2+126x_1x_2^2x_5+14406x_1x_
{-3} | 7x_1^2+3x_1x_2+3x_1x_5-3x_2^3
------------------------------------------------------------------------
^8x_5-49x_2^7x_5^3-21x_2^7x_5^2+21x_2^6x_5^3-21x_2^5x_5^4+21x_2^4x_5^5+
9-1029x_2^8x_5-49x_2^8+343x_2^7x_5^2+98x_2^7x_5-147x_2^6x_5^2+147x_2^5x
2x_5^5-1029x_1x_2x_5^4+294x_1x_2x_5^3+63x_1x_2x_5^2-201684x_2^9+100842x
------------------------------------------------------------------------
9x_2^2x_5^6+9x_2x_5^7
_5^3-147x_2^4x_5^4+21x_2^4x_5^3+9x_2^3x_5^3-63x_2^2x_5^5+18x_2^2x_5^4-
_2^8x_5+7203x_2^8-33614x_2^7x_5^2-12005x_2^7x_5+343x_2^7+14406x_2^6x_5
------------------------------------------------------------------------
63x_2x_5^6+9x_2x_5^5
^2-1029x_2^6x_5-147x_2^6-14406x_2^5x_5^3+1029x_2^5x_5^2+147x_2^5x_5+63x_
------------------------------------------------------------------------
2^5+14406x_2^4x_5^4-1029x_2^4x_5^3+294x_2^4x_5^2+63x_2^4x_5+27x_2^4+189x
------------------------------------------------------------------------
_2^3x_5^2+81x_2^3x_5+6174x_2^2x_5^5-441x_2^2x_5^4+315x_2^2x_5^3+81x_2^2x
------------------------------------------------------------------------
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_5^2+6174x_2x_5^6-441x_2x_5^5+126x_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
7 9 5 9 2 9
o13 = (map(R,R,{-x + --x + x , x , x + -x + x , x }), ideal (-x + --x x
2 1 10 2 4 1 1 3 2 3 2 2 1 10 1 2
-----------------------------------------------------------------------
7 3 101 2 2 3 3 7 2 9 2 2
+ x x + 1, -x x + ---x x + -x x + -x x x + --x x x + x x x +
1 4 2 1 2 15 1 2 2 1 2 2 1 2 3 10 1 2 3 1 2 4
-----------------------------------------------------------------------
5 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
5 8 2 9 7 2 8
o16 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (-x + -x x
2 1 5 2 4 1 3 1 5 2 3 2 2 1 5 1 2
-----------------------------------------------------------------------
5 3 167 2 2 72 3 5 2 8 2 2 2
+ x x + 1, -x x + ---x x + --x x + -x x x + -x x x + -x x x +
1 4 3 1 2 30 1 2 25 1 2 2 1 2 3 5 1 2 3 3 1 2 4
-----------------------------------------------------------------------
9 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
2
o19 = (map(R,R,{x - x + x , x , x + x , x }), ideal (2x - x x + x x +
1 2 4 1 2 3 2 1 1 2 1 4
-----------------------------------------------------------------------
2 2 3 2 2 2
1, x x - x x + x x x - x x x + x x x + x x x x + 1), {x , x })
1 2 1 2 1 2 3 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.