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
5 2 7 12 2 2
o3 = (map(R,R,{-x + -x + x , x , 5x + --x + x , x }), ideal (--x + -x x
7 1 3 2 4 1 1 10 2 3 2 7 1 3 1 2
------------------------------------------------------------------------
25 3 23 2 2 7 3 5 2 2 2 2
+ x x + 1, --x x + --x x + --x x + -x x x + -x x x + 5x x x +
1 4 7 1 2 6 1 2 15 1 2 7 1 2 3 3 1 2 3 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
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 7 7
o6 = (map(R,R,{-x + x + x , x , 9x + -x + x , x + -x + x , x }), ideal
7 1 2 5 1 1 6 2 4 1 5 2 3 2
------------------------------------------------------------------------
2 2 3 8 3 12 2 2 12 2 6 3 12 2
(-x + x x + x x - x , ---x x + --x x + --x x x + -x x + --x x x
7 1 1 2 1 5 2 343 1 2 49 1 2 49 1 2 5 7 1 2 7 1 2 5
------------------------------------------------------------------------
6 2 4 3 2 2 3
+ -x x x + x + 3x x + 3x x + x x ), {x , x , x })
7 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} | 14x_1x_2x_5^6-24x_2^9x_5-14x_2^9+12x_2^8x_5^2+14x_2^8x_5-4x_2^
{-9} | 98x_1x_2^2x_5^3-84x_1x_2x_5^5+98x_1x_2x_5^4+144x_2^9-72x_2^8x_
{-9} | 4802x_1x_2^3+4116x_1x_2^2x_5^2+9604x_1x_2^2x_5+2016x_1x_2x_5^5
{-3} | 2x_1^2+7x_1x_2+7x_1x_5-7x_2^3
------------------------------------------------------------------------
7x_5^3-14x_2^7x_5^2+14x_2^6x_5^3-14x_2^5x_5^4+14x_2^4x_5^5+49x_2^2x_5^6+
5-28x_2^8+24x_2^7x_5^2+56x_2^7x_5-84x_2^6x_5^2+84x_2^5x_5^3-84x_2^4x_5^4
-1176x_1x_2x_5^4+2744x_1x_2x_5^3+4802x_1x_2x_5^2-3456x_2^9+1728x_2^8x_5+
------------------------------------------------------------------------
49x_2x_5^7
+98x_2^4x_5^3+343x_2^3x_5^3-294x_2^2x_5^5+686x_2^2x_5^4-294x_2x_5^6+343x
1008x_2^8-576x_2^7x_5^2-1680x_2^7x_5+392x_2^7+2016x_2^6x_5^2-1176x_2^6x_
------------------------------------------------------------------------
_2x_5^5
5-1372x_2^6-2016x_2^5x_5^3+1176x_2^5x_5^2+1372x_2^5x_5+4802x_2^5+2016x_2
------------------------------------------------------------------------
^4x_5^4-1176x_2^4x_5^3+2744x_2^4x_5^2+4802x_2^4x_5+16807x_2^4+14406x_2^
------------------------------------------------------------------------
3x_5^2+50421x_2^3x_5+7056x_2^2x_5^5-4116x_2^2x_5^4+24010x_2^2x_5^3+
------------------------------------------------------------------------
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50421x_2^2x_5^2+7056x_2x_5^6-4116x_2x_5^5+9604x_2x_5^4+16807x_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
5 4 13 2
o13 = (map(R,R,{-x + x + x , x , -x + 2x + x , x }), ideal (--x + x x +
8 1 2 4 1 3 1 2 3 2 8 1 1 2
-----------------------------------------------------------------------
5 3 31 2 2 3 5 2 2 4 2 2
x x + 1, -x x + --x x + 2x x + -x x x + x x x + -x x x + 2x x x
1 4 6 1 2 12 1 2 1 2 8 1 2 3 1 2 3 3 1 2 4 1 2 4
-----------------------------------------------------------------------
+ x x x x + 1), {x , x })
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 2
o16 = (map(R,R,{10x + x + x , x , x + -x + x , x }), ideal (11x + x x +
1 2 4 1 1 4 2 3 2 1 1 2
-----------------------------------------------------------------------
3 7 2 2 1 3 2 2 2 1 2
x x + 1, 10x x + -x x + -x x + 10x x x + x x x + x x x + -x x x
1 4 1 2 2 1 2 4 1 2 1 2 3 1 2 3 1 2 4 4 1 2 4
-----------------------------------------------------------------------
+ x x x x + 1), {x , x })
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,{- 2x + 2x + x , x , 6x - 4x + x , x }), ideal (- x +
1 2 4 1 1 2 3 2 1
-----------------------------------------------------------------------
3 2 2 3 2 2
2x x + x x + 1, - 12x x + 20x x - 8x x - 2x x x + 2x x x +
1 2 1 4 1 2 1 2 1 2 1 2 3 1 2 3
-----------------------------------------------------------------------
2 2
6x x x - 4x x x + x x x x + 1), {x , x })
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.