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 9 9 2
o3 = (map(R,R,{-x + 7x + x , x , 5x + -x + x , x }), ideal (-x + 7x x +
5 1 2 4 1 1 2 2 3 2 5 1 1 2
------------------------------------------------------------------------
3 193 2 2 63 3 4 2 2 2
x x + 1, 4x x + ---x x + --x x + -x x x + 7x x x + 5x x x +
1 4 1 2 5 1 2 2 1 2 5 1 2 3 1 2 3 1 2 4
------------------------------------------------------------------------
9 2
-x x x + x x x x + 1), {x , x })
2 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
1 5
o6 = (map(R,R,{10x + 3x + x , x , x + -x + x , 4x + -x + x , x }),
1 2 5 1 1 2 2 4 1 6 2 3 2
------------------------------------------------------------------------
2 3 3 2 2 2
ideal (10x + 3x x + x x - x , 1000x x + 900x x + 300x x x +
1 1 2 1 5 2 1 2 1 2 1 2 5
------------------------------------------------------------------------
3 2 2 4 3 2 2 3
270x x + 180x x x + 30x x x + 27x + 27x x + 9x x + x x ), {x , x ,
1 2 1 2 5 1 2 5 2 2 5 2 5 2 5 5 4
------------------------------------------------------------------------
x })
3
o6 : Sequence
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i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 10x_1x_2x_5^6-5400x_2^9x_5-2430x_2^9+900x_2^8x_5^2+810x_2^8x_5
{-9} | 270x_1x_2^2x_5^3-100x_1x_2x_5^5+90x_1x_2x_5^4+54000x_2^9-9000x
{-9} | 196830x_1x_2^3+72900x_1x_2^2x_5^2+131220x_1x_2^2x_5+20000x_1x_
{-3} | 10x_1^2+3x_1x_2+x_1x_5-x_2^3
------------------------------------------------------------------------
-100x_2^7x_5^3-270x_2^7x_5^2+90x_2^6x_5^3-30x_2^5x_5^4+10x_2^4x_5^5+3x_
_2^8x_5-2700x_2^8+1000x_2^7x_5^2+1800x_2^7x_5-900x_2^6x_5^2+300x_2^5x_5
2x_5^5-9000x_1x_2x_5^4+16200x_1x_2x_5^3+21870x_1x_2x_5^2-10800000x_2^9+
------------------------------------------------------------------------
2^2x_5^6+x_2x_5^7
^3-100x_2^4x_5^4+90x_2^4x_5^3+81x_2^3x_5^3-30x_2^2x_5^5+54x_2^2x_5^4-
1800000x_2^8x_5+810000x_2^8-200000x_2^7x_5^2-450000x_2^7x_5+81000x_2^
------------------------------------------------------------------------
10x_2x_5^6+9x_2x_5^5
7+180000x_2^6x_5^2-81000x_2^6x_5-72900x_2^6-60000x_2^5x_5^3+27000x_2^5x_
------------------------------------------------------------------------
5^2+24300x_2^5x_5+65610x_2^5+20000x_2^4x_5^4-9000x_2^4x_5^3+16200x_2^4x_
------------------------------------------------------------------------
5^2+21870x_2^4x_5+59049x_2^4+21870x_2^3x_5^2+59049x_2^3x_5+6000x_2^2x_5^
------------------------------------------------------------------------
5-2700x_2^2x_5^4+12150x_2^2x_5^3+19683x_2^2x_5^2+2000x_2x_5^6-900x_2x_5^
------------------------------------------------------------------------
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5+1620x_2x_5^4+2187x_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
|
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
3 1 3 2 3
o13 = (map(R,R,{2x + -x + x , x , -x + --x + x , x }), ideal (3x + -x x
1 2 2 4 1 7 1 10 2 3 2 1 2 1 2
-----------------------------------------------------------------------
2 3 57 2 2 9 3 2 3 2 1 2
+ x x + 1, -x x + --x x + --x x + 2x x x + -x x x + -x x x +
1 4 7 1 2 70 1 2 20 1 2 1 2 3 2 1 2 3 7 1 2 4
-----------------------------------------------------------------------
3 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
3 7 3 1 13 2
o16 = (map(R,R,{--x + -x + x , x , -x + -x + x , x }), ideal (--x +
10 1 8 2 4 1 8 1 3 2 3 2 10 1
-----------------------------------------------------------------------
7 9 3 137 2 2 7 3 3 2 7 2
-x x + x x + 1, --x x + ---x x + --x x + --x x x + -x x x +
8 1 2 1 4 80 1 2 320 1 2 24 1 2 10 1 2 3 8 1 2 3
-----------------------------------------------------------------------
3 2 1 2
-x x x + -x x x + x x x x + 1), {x , x })
8 1 2 4 3 1 2 4 1 2 3 4 4 3
o16 : Sequence
|
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 + 2x + x , x , 2x + x , x }), ideal (2x + 2x x + x x
1 2 4 1 2 3 2 1 1 2 1 4
-----------------------------------------------------------------------
2 2 3 2 2 2
+ 1, 2x x + 4x x + x x x + 2x x x + 2x x x + x x x x + 1), {x ,
1 2 1 2 1 2 3 1 2 3 1 2 4 1 2 3 4 4
-----------------------------------------------------------------------
x })
3
o19 : Sequence
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This symbol is provided by the package NoetherNormalization.