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
7 1 5 2 7
o3 = (map(R,R,{x + -x + x , x , -x + -x + x , x }), ideal (2x + -x x +
1 4 2 4 1 6 1 4 2 3 2 1 4 1 2
------------------------------------------------------------------------
1 3 37 2 2 35 3 2 7 2 1 2 5 2
x x + 1, -x x + --x x + --x x + x x x + -x x x + -x x x + -x x x
1 4 6 1 2 24 1 2 16 1 2 1 2 3 4 1 2 3 6 1 2 4 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
5 2 3
o6 = (map(R,R,{-x + x + x , x , -x + x + x , x + -x + x , x }), ideal
2 1 2 5 1 3 1 2 4 1 2 2 3 2
------------------------------------------------------------------------
5 2 3 125 3 75 2 2 75 2 15 3 2
(-x + x x + x x - x , ---x x + --x x + --x x x + --x x + 15x 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
------------------------------------------------------------------------
15 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} | 10x_1x_2x_5^6-150x_2^9x_5-10x_2^9+75x_2^8x_5^2+10x_2^8x_5-25x_2
{-9} | 20x_1x_2^2x_5^3-150x_1x_2x_5^5+20x_1x_2x_5^4+2250x_2^9-1125x_2^
{-9} | 40x_1x_2^3+300x_1x_2^2x_5^2+80x_1x_2^2x_5+11250x_1x_2x_5^5-750x
{-3} | 5x_1^2+2x_1x_2+2x_1x_5-2x_2^3
------------------------------------------------------------------------
^7x_5^3-10x_2^7x_5^2+10x_2^6x_5^3-10x_2^5x_5^4+10x_2^4x_5^5+4x_2^2x_5^6+
8x_5-50x_2^8+375x_2^7x_5^2+100x_2^7x_5-150x_2^6x_5^2+150x_2^5x_5^3-150x_
_1x_2x_5^4+200x_1x_2x_5^3+40x_1x_2x_5^2-168750x_2^9+84375x_2^8x_5+5625x_
------------------------------------------------------------------------
4x_2x_5^7
2^4x_5^4+20x_2^4x_5^3+8x_2^3x_5^3-60x_2^2x_5^5+16x_2^2x_5^4-60x_2x_5^6
2^8-28125x_2^7x_5^2-9375x_2^7x_5+250x_2^7+11250x_2^6x_5^2-750x_2^6x_5-
------------------------------------------------------------------------
+8x_2x_5^5
100x_2^6-11250x_2^5x_5^3+750x_2^5x_5^2+100x_2^5x_5+40x_2^5+11250x_2^4x_5
------------------------------------------------------------------------
^4-750x_2^4x_5^3+200x_2^4x_5^2+40x_2^4x_5+16x_2^4+120x_2^3x_5^2+48x_2^3x
------------------------------------------------------------------------
_5+4500x_2^2x_5^5-300x_2^2x_5^4+200x_2^2x_5^3+48x_2^2x_5^2+4500x_2x_5^6-
------------------------------------------------------------------------
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300x_2x_5^5+80x_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
4 9 10 13 2
o13 = (map(R,R,{-x + 9x + x , x , -x + --x + x , x }), ideal (--x +
9 1 2 4 1 7 1 9 2 3 2 9 1
-----------------------------------------------------------------------
4 3 6841 2 2 3 4 2 2
9x x + x x + 1, -x x + ----x x + 10x x + -x x x + 9x x x +
1 2 1 4 7 1 2 567 1 2 1 2 9 1 2 3 1 2 3
-----------------------------------------------------------------------
9 2 10 2
-x x x + --x x x + x x x x + 1), {x , x })
7 1 2 4 9 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
1 5 2 1
o16 = (map(R,R,{7x + -x + x , x , 4x + -x + x , x }), ideal (8x + -x x
1 4 2 4 1 1 4 2 3 2 1 4 1 2
-----------------------------------------------------------------------
3 39 2 2 5 3 2 1 2 2
+ x x + 1, 28x x + --x x + --x x + 7x x x + -x x x + 4x x x +
1 4 1 2 4 1 2 16 1 2 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
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,{x + 2x + x , x , 2x + 6x + x , x }), ideal (2x + 2x x +
1 2 4 1 1 2 3 2 1 1 2
-----------------------------------------------------------------------
3 2 2 3 2 2 2
x x + 1, 2x x + 10x x + 12x x + x x x + 2x x x + 2x x x +
1 4 1 2 1 2 1 2 1 2 3 1 2 3 1 2 4
-----------------------------------------------------------------------
2
6x 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.