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 2
o3 = (map(R,R,{x + 4x + x , x , -x + 3x + x , x }), ideal (2x + 4x x +
1 2 4 1 9 1 2 3 2 1 1 2
------------------------------------------------------------------------
7 3 55 2 2 3 2 2 7 2 2
x x + 1, -x x + --x x + 12x x + x x x + 4x x x + -x x x + 3x x x
1 4 9 1 2 9 1 2 1 2 1 2 3 1 2 3 9 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
7 1 2 2
o6 = (map(R,R,{-x + -x + x , x , -x + x + x , 5x + -x + x , x }), ideal
4 1 2 2 5 1 3 1 2 4 1 9 2 3 2
------------------------------------------------------------------------
7 2 1 3 343 3 147 2 2 147 2 21 3
(-x + -x x + x x - x , ---x x + ---x x + ---x x x + --x x +
4 1 2 1 2 1 5 2 64 1 2 32 1 2 16 1 2 5 16 1 2
------------------------------------------------------------------------
21 2 21 2 1 4 3 3 3 2 2 3
--x x x + --x x x + -x + -x x + -x x + x x ), {x , x , x })
4 1 2 5 4 1 2 5 8 2 4 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} | 224x_1x_2x_5^6-588x_2^9x_5-7x_2^9+588x_2^8x_5^2+14x_2^8x
{-9} | 28x_1x_2^2x_5^3-2352x_1x_2x_5^5+56x_1x_2x_5^4+6174x_2^9-
{-9} | 7x_1x_2^3+588x_1x_2^2x_5^2+28x_1x_2^2x_5+1382976x_1x_2x_
{-3} | 7x_1^2+2x_1x_2+4x_1x_5-4x_2^3
------------------------------------------------------------------------
_5-392x_2^7x_5^3-28x_2^7x_5^2+56x_2^6x_5^3-112x_2^5x_5^4+224x_
6174x_2^8x_5-49x_2^8+4116x_2^7x_5^2+196x_2^7x_5-588x_2^6x_5^2+
5^5-16464x_1x_2x_5^4+784x_1x_2x_5^3+28x_1x_2x_5^2-3630312x_2^9
------------------------------------------------------------------------
2^4x_5^5+64x_2^2x_5^6+128x_2x_5^7
1176x_2^5x_5^3-2352x_2^4x_5^4+56x_2^4x_5^3+8x_2^3x_5^3-672x_2^2x_5^5+
+3630312x_2^8x_5+43218x_2^8-2420208x_2^7x_5^2-144060x_2^7x_5+686x_2^7
------------------------------------------------------------------------
32x_2^2x_5^4-1344x_2x_5^6+32x_2x_5^5
+345744x_2^6x_5^2-4116x_2^6x_5-98x_2^6-691488x_2^5x_5^3+8232x_2^5x_5^2+
------------------------------------------------------------------------
196x_2^5x_5+14x_2^5+1382976x_2^4x_5^4-16464x_2^4x_5^3+784x_2^4x_5^2+28x_
------------------------------------------------------------------------
2^4x_5+2x_2^4+168x_2^3x_5^2+12x_2^3x_5+395136x_2^2x_5^5-4704x_2^2x_5^4+
------------------------------------------------------------------------
560x_2^2x_5^3+24x_2^2x_5^2+790272x_2x_5^6-9408x_2x_5^5+448x_2x_5^4+16x_
------------------------------------------------------------------------
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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
2 1 3 10 9 2 1
o13 = (map(R,R,{-x + -x + x , x , -x + --x + x , x }), ideal (-x + -x x
7 1 9 2 4 1 2 1 7 2 3 2 7 1 9 1 2
-----------------------------------------------------------------------
3 3 169 2 2 10 3 2 2 1 2 3 2
+ x x + 1, -x x + ---x x + --x x + -x x x + -x x x + -x x x +
1 4 7 1 2 294 1 2 63 1 2 7 1 2 3 9 1 2 3 2 1 2 4
-----------------------------------------------------------------------
10 2
--x x x + x x x x + 1), {x , x })
7 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 5 8 2 5
o16 = (map(R,R,{-x + -x + x , x , 6x + -x + x , x }), ideal (-x + -x x
7 1 2 2 4 1 1 2 2 3 2 7 1 2 1 2
-----------------------------------------------------------------------
6 3 215 2 2 25 3 1 2 5 2 2
+ x x + 1, -x x + ---x x + --x x + -x x x + -x x x + 6x x x +
1 4 7 1 2 14 1 2 4 1 2 7 1 2 3 2 1 2 3 1 2 4
-----------------------------------------------------------------------
5 2
-x x x + x x x x + 1), {x , x })
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
2
o19 = (map(R,R,{- 4x + x + x , x , - 4x - 6x + x , x }), ideal (- 3x +
1 2 4 1 1 2 3 2 1
-----------------------------------------------------------------------
3 2 2 3 2 2 2
x x + x x + 1, 16x x + 20x x - 6x x - 4x x x + x x x - 4x x x -
1 2 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.