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
1 1 8 5 2 1
o3 = (map(R,R,{-x + -x + x , x , x + -x + x , x }), ideal (-x + -x x +
4 1 9 2 4 1 1 5 2 3 2 4 1 9 1 2
------------------------------------------------------------------------
1 3 23 2 2 8 3 1 2 1 2 2 8 2
x x + 1, -x x + --x x + --x x + -x x x + -x x x + x x x + -x x x
1 4 4 1 2 45 1 2 45 1 2 4 1 2 3 9 1 2 3 1 2 4 5 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
1 4 9
o6 = (map(R,R,{5x + 3x + x , x , 2x + -x + x , -x + -x + x , x }),
1 2 5 1 1 5 2 4 5 1 2 2 3 2
------------------------------------------------------------------------
2 3 3 2 2 2 3
ideal (5x + 3x x + x x - x , 125x x + 225x x + 75x x x + 135x x +
1 1 2 1 5 2 1 2 1 2 1 2 5 1 2
------------------------------------------------------------------------
2 2 4 3 2 2 3
90x x x + 15x x x + 27x + 27x x + 9x x + x x ), {x , x , x })
1 2 5 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} | 5x_1x_2x_5^6-1350x_2^9x_5-1215x_2^9+225x_2^8x_5^2+405x_2^8x_5-
{-9} | 135x_1x_2^2x_5^3-25x_1x_2x_5^5+45x_1x_2x_5^4+6750x_2^9-1125x_2
{-9} | 98415x_1x_2^3+18225x_1x_2^2x_5^2+65610x_1x_2^2x_5+1250x_1x_2x_
{-3} | 5x_1^2+3x_1x_2+x_1x_5-x_2^3
------------------------------------------------------------------------
25x_2^7x_5^3-135x_2^7x_5^2+45x_2^6x_5^3-15x_2^5x_5^4+5x_2^4x_5^5+3x_2^2x
^8x_5-675x_2^8+125x_2^7x_5^2+450x_2^7x_5-225x_2^6x_5^2+75x_2^5x_5^3-25x_
5^5-1125x_1x_2x_5^4+4050x_1x_2x_5^3+10935x_1x_2x_5^2-337500x_2^9+56250x_
------------------------------------------------------------------------
_5^6+x_2x_5^7
2^4x_5^4+45x_2^4x_5^3+81x_2^3x_5^3-15x_2^2x_5^5+54x_2^2x_5^4-5x_2x_5^6+
2^8x_5+50625x_2^8-6250x_2^7x_5^2-28125x_2^7x_5+10125x_2^7+11250x_2^6x_5
------------------------------------------------------------------------
9x_2x_5^5
^2-10125x_2^6x_5-18225x_2^6-3750x_2^5x_5^3+3375x_2^5x_5^2+6075x_2^5x_5+
------------------------------------------------------------------------
32805x_2^5+1250x_2^4x_5^4-1125x_2^4x_5^3+4050x_2^4x_5^2+10935x_2^4x_5+
------------------------------------------------------------------------
59049x_2^4+10935x_2^3x_5^2+59049x_2^3x_5+750x_2^2x_5^5-675x_2^2x_5^4+
------------------------------------------------------------------------
6075x_2^2x_5^3+19683x_2^2x_5^2+250x_2x_5^6-225x_2x_5^5+810x_2x_5^4+2187x
------------------------------------------------------------------------
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_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 1 5 12 2 1
o13 = (map(R,R,{-x + -x + x , x , x + -x + x , x }), ideal (--x + -x x
5 1 4 2 4 1 1 4 2 3 2 5 1 4 1 2
-----------------------------------------------------------------------
7 3 2 2 5 3 7 2 1 2 2
+ x x + 1, -x x + 2x x + --x x + -x x x + -x x x + x x x +
1 4 5 1 2 1 2 16 1 2 5 1 2 3 4 1 2 3 1 2 4
-----------------------------------------------------------------------
5 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
10 3 3 17 2
o16 = (map(R,R,{--x + -x + x , x , 4x + -x + x , x }), ideal (--x +
7 1 5 2 4 1 1 2 2 3 2 7 1
-----------------------------------------------------------------------
3 40 3 159 2 2 9 3 10 2 3 2
-x x + x x + 1, --x x + ---x x + --x x + --x x x + -x x x +
5 1 2 1 4 7 1 2 35 1 2 10 1 2 7 1 2 3 5 1 2 3
-----------------------------------------------------------------------
2 3 2
4x x x + -x x x + x x x x + 1), {x , x })
1 2 4 2 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
2
o19 = (map(R,R,{3x + x , x , - x + x , x }), ideal (x + 3x x + x x + 1,
2 4 1 2 3 2 1 1 2 1 4
-----------------------------------------------------------------------
3 2 2
- 3x x + 3x x x - x 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.