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
2 6 6 7 2
o3 = (map(R,R,{-x + x + x , x , -x + -x + x , x }), ideal (-x + x x +
5 1 2 4 1 5 1 5 2 3 2 5 1 1 2
------------------------------------------------------------------------
12 3 42 2 2 6 3 2 2 2 6 2 6 2
x x + 1, --x x + --x x + -x x + -x x x + x x x + -x x x + -x x x
1 4 25 1 2 25 1 2 5 1 2 5 1 2 3 1 2 3 5 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
7 3 5
o6 = (map(R,R,{x + -x + x , x , -x + 2x + x , -x + 3x + x , x }), ideal
1 2 2 5 1 7 1 2 4 9 1 2 3 2
------------------------------------------------------------------------
2 7 3 3 21 2 2 2 147 3 2
(x + -x x + x x - x , x x + --x x + 3x x x + ---x x + 21x x x +
1 2 1 2 1 5 2 1 2 2 1 2 1 2 5 4 1 2 1 2 5
------------------------------------------------------------------------
2 343 4 147 3 21 2 2 3
3x x x + ---x + ---x x + --x x + x x ), {x , x , x })
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} | 32x_1x_2x_5^6-2352x_2^9x_5-16807x_2^9+336x_2^8x_5^2+4802x_2^8x_
{-9} | 4802x_1x_2^2x_5^3-96x_1x_2x_5^5+1372x_1x_2x_5^4+7056x_2^9-1008x
{-9} | 3954653486x_1x_2^3+79060128x_1x_2^2x_5^2+2259801992x_1x_2^2x_5+
{-3} | 2x_1^2+7x_1x_2+2x_1x_5-2x_2^3
------------------------------------------------------------------------
5-32x_2^7x_5^3-1372x_2^7x_5^2+392x_2^6x_5^3-112x_2^5x_5^4+32x_2^4x_5^5+
_2^8x_5-4802x_2^8+96x_2^7x_5^2+2744x_2^7x_5-1176x_2^6x_5^2+336x_2^5x_5^
73728x_1x_2x_5^5-526848x_1x_2x_5^4+15059072x_1x_2x_5^3+322828856x_1x_2x
------------------------------------------------------------------------
112x_2^2x_5^6+32x_2x_5^7
3-96x_2^4x_5^4+1372x_2^4x_5^3+16807x_2^3x_5^3-336x_2^2x_5^5+9604x_2^2x_5
_5^2-5419008x_2^9+774144x_2^8x_5+5531904x_2^8-73728x_2^7x_5^2-2634240x_2
------------------------------------------------------------------------
^4-96x_2x_5^6+1372x_2x_5^5
^7x_5+7529536x_2^7+903168x_2^6x_5^2-6453888x_2^6x_5-92236816x_2^6-
------------------------------------------------------------------------
258048x_2^5x_5^3+1843968x_2^5x_5^2+26353376x_2^5x_5+1129900996x_2^5+
------------------------------------------------------------------------
73728x_2^4x_5^4-526848x_2^4x_5^3+15059072x_2^4x_5^2+322828856x_2^4x_5+
------------------------------------------------------------------------
13841287201x_2^4+276710448x_2^3x_5^2+11863960458x_2^3x_5+258048x_2^2x_5^
------------------------------------------------------------------------
5-1843968x_2^2x_5^4+131766880x_2^2x_5^3+3389702988x_2^2x_5^2+73728x_2x_5
------------------------------------------------------------------------
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^6-526848x_2x_5^5+15059072x_2x_5^4+322828856x_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
7 1 8 11 2
o13 = (map(R,R,{-x + x + x , x , -x + -x + x , x }), ideal (--x + x x +
4 1 2 4 1 5 1 7 2 3 2 4 1 1 2
-----------------------------------------------------------------------
7 3 11 2 2 8 3 7 2 2 1 2
x x + 1, --x x + --x x + -x x + -x x x + x x x + -x x x +
1 4 20 1 2 5 1 2 7 1 2 4 1 2 3 1 2 3 5 1 2 4
-----------------------------------------------------------------------
8 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
2 3 5 11 2 3
o16 = (map(R,R,{-x + -x + x , x , 4x + -x + x , x }), ideal (--x + -x x
9 1 4 2 4 1 1 4 2 3 2 9 1 4 1 2
-----------------------------------------------------------------------
8 3 59 2 2 15 3 2 2 3 2 2
+ x x + 1, -x x + --x x + --x x + -x x x + -x x x + 4x x x +
1 4 9 1 2 18 1 2 16 1 2 9 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
2
o19 = (map(R,R,{- 2x + x , x , - 2x + 2x + x , x }), ideal (x - 2x x +
2 4 1 1 2 3 2 1 1 2
-----------------------------------------------------------------------
2 2 3 2 2 2
x x + 1, 4x x - 4x x - 2x x x - 2x x x + 2x x x + x x x x + 1),
1 4 1 2 1 2 1 2 3 1 2 4 1 2 4 1 2 3 4
-----------------------------------------------------------------------
{x , x })
4 3
o19 : Sequence
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This symbol is provided by the package NoetherNormalization.