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
3 3 3 1 5 2 3
o3 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (-x + -x x +
2 1 4 2 4 1 4 1 9 2 3 2 2 1 4 1 2
------------------------------------------------------------------------
9 3 35 2 2 1 3 3 2 3 2 3 2
x x + 1, -x x + --x x + --x x + -x x x + -x x x + -x x x +
1 4 8 1 2 48 1 2 12 1 2 2 1 2 3 4 1 2 3 4 1 2 4
------------------------------------------------------------------------
1 2
-x x x + x x x x + 1), {x , x })
9 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
3 1 4 3 4
o6 = (map(R,R,{4x + -x + x , x , -x + -x + x , --x + -x + x , x }),
1 4 2 5 1 3 1 3 2 4 10 1 5 2 3 2
------------------------------------------------------------------------
2 3 3 3 2 2 2 27 3
ideal (4x + -x x + x x - x , 64x x + 36x x + 48x x x + --x x +
1 4 1 2 1 5 2 1 2 1 2 1 2 5 4 1 2
------------------------------------------------------------------------
2 2 27 4 27 3 9 2 2 3
18x x x + 12x x x + --x + --x x + -x x + x x ), {x , x , x })
1 2 5 1 2 5 64 2 16 2 5 4 2 5 2 5 5 4 3
o6 : Sequence
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i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 1024x_1x_2x_5^6-13824x_2^9x_5-243x_2^9+9216x_2^8x
{-9} | 432x_1x_2^2x_5^3-16384x_1x_2x_5^5+576x_1x_2x_5^4+
{-9} | 314928x_1x_2^3+11943936x_1x_2^2x_5^2+839808x_1x_2
{-3} | 16x_1^2+3x_1x_2+4x_1x_5-4x_2^3
------------------------------------------------------------------------
_5^2+324x_2^8x_5-4096x_2^7x_5^3-432x_2^7x_5^2+576x_2^6x_5^3-768x_2
221184x_2^9-147456x_2^8x_5-1728x_2^8+65536x_2^7x_5^2+4608x_2^7x_5-
^2x_5+8589934592x_1x_2x_5^5-150994944x_1x_2x_5^4+10616832x_1x_2x_5
------------------------------------------------------------------------
^5x_5^4+1024x_2^4x_5^5+192x_2^2x_5^6+256x_2x_5^7
9216x_2^6x_5^2+12288x_2^5x_5^3-16384x_2^4x_5^4+576x_2^4x_5^3+81x_2^3x_5^
^3+559872x_1x_2x_5^2-115964116992x_2^9+77309411328x_2^8x_5+1358954496x_2
------------------------------------------------------------------------
3-3072x_2^2x_5^5+216x_2^2x_5^4-4096x_2x_5^6+144x_2x_5^5
^8-34359738368x_2^7x_5^2-3019898880x_2^7x_5+21233664x_2^7+4831838208x_2^
------------------------------------------------------------------------
6x_5^2-84934656x_2^6x_5-2985984x_2^6-6442450944x_2^5x_5^3+113246208x_2^
------------------------------------------------------------------------
5x_5^2+3981312x_2^5x_5+419904x_2^5+8589934592x_2^4x_5^4-150994944x_2^4x_
------------------------------------------------------------------------
5^3+10616832x_2^4x_5^2+559872x_2^4x_5+59049x_2^4+2239488x_2^3x_5^2+
------------------------------------------------------------------------
236196x_2^3x_5+1610612736x_2^2x_5^5-28311552x_2^2x_5^4+4976640x_2^2x_5^3
------------------------------------------------------------------------
+314928x_2^2x_5^2+2147483648x_2x_5^6-37748736x_2x_5^5+2654208x_2x_5^4+
------------------------------------------------------------------------
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139968x_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
1 9 1 7 4 2 9
o13 = (map(R,R,{-x + -x + x , x , -x + --x + x , x }), ideal (-x + -x x
3 1 4 2 4 1 5 1 10 2 3 2 3 1 4 1 2
-----------------------------------------------------------------------
1 3 41 2 2 63 3 1 2 9 2 1 2
+ x x + 1, --x x + --x x + --x x + -x x x + -x x x + -x x x +
1 4 15 1 2 60 1 2 40 1 2 3 1 2 3 4 1 2 3 5 1 2 4
-----------------------------------------------------------------------
7 2
--x x x + x x x x + 1), {x , x })
10 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
7 1 5 11 2 1
o16 = (map(R,R,{-x + -x + x , x , 3x + -x + x , x }), ideal (--x + -x x
4 1 4 2 4 1 1 3 2 3 2 4 1 4 1 2
-----------------------------------------------------------------------
21 3 11 2 2 5 3 7 2 1 2 2
+ x x + 1, --x x + --x x + --x x + -x x x + -x x x + 3x x x +
1 4 4 1 2 3 1 2 12 1 2 4 1 2 3 4 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
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 + 4x + x , x , - 5x - 4x + x , x }), ideal (- 3x +
1 2 4 1 1 2 3 2 1
-----------------------------------------------------------------------
3 2 2 3 2 2 2
4x x + x x + 1, 20x x - 4x x - 16x x - 4x x x + 4x x x - 5x x x
1 2 1 4 1 2 1 2 1 2 1 2 3 1 2 3 1 2 4
-----------------------------------------------------------------------
2
- 4x 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.