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
4 9 2 4
o3 = (map(R,R,{x + -x + x , x , --x + x + x , x }), ideal (2x + -x x +
1 5 2 4 1 10 1 2 3 2 1 5 1 2
------------------------------------------------------------------------
9 3 43 2 2 4 3 2 4 2 9 2 2
x x + 1, --x x + --x x + -x x + x x x + -x x x + --x x x + x x x
1 4 10 1 2 25 1 2 5 1 2 1 2 3 5 1 2 3 10 1 2 4 1 2 4
------------------------------------------------------------------------
+ x x x x + 1), {x , x })
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
4 1 3 5 1 5
o6 = (map(R,R,{-x + -x + x , x , -x + -x + x , -x + -x + x , x }),
7 1 6 2 5 1 2 1 8 2 4 5 1 4 2 3 2
------------------------------------------------------------------------
4 2 1 3 64 3 8 2 2 48 2 1 3
ideal (-x + -x x + x x - x , ---x x + --x x + --x x x + --x x +
7 1 6 1 2 1 5 2 343 1 2 49 1 2 49 1 2 5 21 1 2
------------------------------------------------------------------------
4 2 12 2 1 4 1 3 1 2 2 3
-x x x + --x x x + ---x + --x x + -x x + x x ), {x , x , x })
7 1 2 5 7 1 2 5 216 2 12 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} | 54432x_1x_2x_5^6-5184x_2^9x_5-7x_2^9+15552x_2^8x_5^2+
{-9} | 1176x_1x_2^2x_5^3-2612736x_1x_2x_5^5+7056x_1x_2x_5^4+
{-9} | 57624x_1x_2^3+128024064x_1x_2^2x_5^2+691488x_1x_2^2x_
{-3} | 24x_1^2+7x_1x_2+42x_1x_5-42x_2^3
------------------------------------------------------------------------
42x_2^8x_5-31104x_2^7x_5^3-252x_2^7x_5^2+1512x_2^6x_5^3-9072x_2^5x_
248832x_2^9-746496x_2^8x_5-672x_2^8+1492992x_2^7x_5^2+8064x_2^7x_5-
5+70214291030016x_1x_2x_5^5-94810963968x_1x_2x_5^4+512096256x_1x_2x
------------------------------------------------------------------------
5^4+54432x_2^4x_5^5+15876x_2^2x_5^6+95256x_2x_5^7
72576x_2^6x_5^2+435456x_2^5x_5^3-2612736x_2^4x_5^4+7056x_2^4x_5^3+
_5^3+2074464x_1x_2x_5^2-6687075336192x_2^9+20061226008576x_2^8x_5+
------------------------------------------------------------------------
343x_2^3x_5^3-762048x_2^2x_5^5+4116x_2^2x_5^4-4572288x_2x_5^6+
27088846848x_2^8-40122452017152x_2^7x_5^2-270888468480x_2^7x_5
------------------------------------------------------------------------
12348x_2x_5^5
+146313216x_2^7+1950396973056x_2^6x_5^2-2633637888x_2^6x_5-7112448x_2^6-
------------------------------------------------------------------------
11702381838336x_2^5x_5^3+15801827328x_2^5x_5^2+42674688x_2^5x_5+345744x_
------------------------------------------------------------------------
2^5+70214291030016x_2^4x_5^4-94810963968x_2^4x_5^3+512096256x_2^4x_5^2+
------------------------------------------------------------------------
2074464x_2^4x_5+16807x_2^4+37340352x_2^3x_5^2+302526x_2^3x_5+
------------------------------------------------------------------------
20479168217088x_2^2x_5^5-27653197824x_2^2x_5^4+373403520x_2^2x_5^3+
------------------------------------------------------------------------
1815156x_2^2x_5^2+122875009302528x_2x_5^6-165919186944x_2x_5^5+
------------------------------------------------------------------------
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896168448x_2x_5^4+3630312x_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
3 7 5 2
o13 = (map(R,R,{-x + x + x , x , -x + 6x + x , x }), ideal (-x + x x +
2 1 2 4 1 2 1 2 3 2 2 1 1 2
-----------------------------------------------------------------------
21 3 25 2 2 3 3 2 2 7 2
x x + 1, --x x + --x x + 6x x + -x x x + x x x + -x x x +
1 4 4 1 2 2 1 2 1 2 2 1 2 3 1 2 3 2 1 2 4
-----------------------------------------------------------------------
2
6x 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
3 8 1 2
o16 = (map(R,R,{2x + --x + x , x , -x + -x + x , x }), ideal (3x +
1 10 2 4 1 7 1 3 2 3 2 1
-----------------------------------------------------------------------
3 16 3 106 2 2 1 3 2 3 2
--x x + x x + 1, --x x + ---x x + --x x + 2x x x + --x x x +
10 1 2 1 4 7 1 2 105 1 2 10 1 2 1 2 3 10 1 2 3
-----------------------------------------------------------------------
8 2 1 2
-x x x + -x x x + x x x x + 1), {x , x })
7 1 2 4 3 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
--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: 3
--trying with basis element limit: 5
--trying with basis element limit: 20
2
o19 = (map(R,R,{6x - x + x , x , - 6x - x + x , x }), ideal (7x - x x +
1 2 4 1 1 2 3 2 1 1 2
-----------------------------------------------------------------------
3 3 2 2 2 2
x x + 1, - 36x x + x x + 6x x x - x x x - 6x x x - x x x +
1 4 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.