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
4 10 1 1 7 2 10
o3 = (map(R,R,{-x + --x + x , x , -x + -x + x , x }), ideal (-x + --x x
3 1 7 2 4 1 5 1 7 2 3 2 3 1 7 1 2
------------------------------------------------------------------------
4 3 10 2 2 10 3 4 2 10 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 21 1 2 49 1 2 3 1 2 3 7 1 2 3 5 1 2 4
------------------------------------------------------------------------
1 2
-x x x + x x x x + 1), {x , x })
7 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
2 3 3 1
o6 = (map(R,R,{x + -x + x , x , -x + -x + x , 3x + -x + x , x }), ideal
1 3 2 5 1 4 1 2 2 4 1 5 2 3 2
------------------------------------------------------------------------
2 2 3 3 2 2 2 4 3 2
(x + -x x + x x - x , x x + 2x x + 3x x x + -x x + 4x x x +
1 3 1 2 1 5 2 1 2 1 2 1 2 5 3 1 2 1 2 5
------------------------------------------------------------------------
2 8 4 4 3 2 2 3
3x x x + --x + -x x + 2x x + x x ), {x , x , x })
1 2 5 27 2 3 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} | 243x_1x_2x_5^6-648x_2^9x_5-32x_2^9+486x_2^8x_5^2+48x_2^8x_5-
{-9} | 48x_1x_2^2x_5^3-729x_1x_2x_5^5+72x_1x_2x_5^4+1944x_2^9-1458x
{-9} | 3072x_1x_2^3+46656x_1x_2^2x_5^2+9216x_1x_2^2x_5+4782969x_1x_
{-3} | 3x_1^2+2x_1x_2+3x_1x_5-3x_2^3
------------------------------------------------------------------------
243x_2^7x_5^3-72x_2^7x_5^2+108x_2^6x_5^3-162x_2^5x_5^4+243x_2^4x_5^5+
_2^8x_5-48x_2^8+729x_2^7x_5^2+144x_2^7x_5-324x_2^6x_5^2+486x_2^5x_5^3
2x_5^5-236196x_1x_2x_5^4+46656x_1x_2x_5^3+6912x_1x_2x_5^2-12754584x_2
------------------------------------------------------------------------
162x_2^2x_5^6+243x_2x_5^7
-729x_2^4x_5^4+72x_2^4x_5^3+32x_2^3x_5^3-486x_2^2x_5^5+96x_2^2x_5^4-729x
^9+9565938x_2^8x_5+472392x_2^8-4782969x_2^7x_5^2-1180980x_2^7x_5+23328x_
------------------------------------------------------------------------
_2x_5^6+72x_2x_5^5
2^7+2125764x_2^6x_5^2-104976x_2^6x_5-10368x_2^6-3188646x_2^5x_5^3+
------------------------------------------------------------------------
157464x_2^5x_5^2+15552x_2^5x_5+4608x_2^5+4782969x_2^4x_5^4-236196x_2^4x_
------------------------------------------------------------------------
5^3+46656x_2^4x_5^2+6912x_2^4x_5+2048x_2^4+31104x_2^3x_5^2+9216x_2^3x_5+
------------------------------------------------------------------------
3188646x_2^2x_5^5-157464x_2^2x_5^4+77760x_2^2x_5^3+13824x_2^2x_5^2+
------------------------------------------------------------------------
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4782969x_2x_5^6-236196x_2x_5^5+46656x_2x_5^4+6912x_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 2 5 7 2
o13 = (map(R,R,{-x + --x + x , x , -x + -x + x , x }), ideal (-x +
5 1 10 2 4 1 5 1 3 2 3 2 5 1
-----------------------------------------------------------------------
1 4 3 53 2 2 1 3 2 2 1 2
--x x + x x + 1, --x x + --x x + -x x + -x x x + --x x x +
10 1 2 1 4 25 1 2 75 1 2 6 1 2 5 1 2 3 10 1 2 3
-----------------------------------------------------------------------
2 2 5 2
-x x x + -x x x + x x x x + 1), {x , x })
5 1 2 4 3 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
3 7 4 8 5 2 7
o16 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (-x + -x x
2 1 5 2 4 1 3 1 3 2 3 2 2 1 5 1 2
-----------------------------------------------------------------------
3 88 2 2 56 3 3 2 7 2 4 2
+ x x + 1, 2x x + --x x + --x x + -x x x + -x x x + -x x x +
1 4 1 2 15 1 2 15 1 2 2 1 2 3 5 1 2 3 3 1 2 4
-----------------------------------------------------------------------
8 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
2
o19 = (map(R,R,{x + x , x , x - 2x + x , x }), ideal (x + x x + x x +
2 4 1 1 2 3 2 1 1 2 1 4
-----------------------------------------------------------------------
2 2 3 2 2 2
1, x x - 2x x + x x x + x x x - 2x x x + x x x x + 1), {x , x })
1 2 1 2 1 2 3 1 2 4 1 2 4 1 2 3 4 4 3
o19 : Sequence
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This symbol is provided by the package NoetherNormalization.