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
7 9 1 10 16 2 9
o3 = (map(R,R,{-x + -x + x , x , -x + --x + x , x }), ideal (--x + -x x
9 1 8 2 4 1 3 1 7 2 3 2 9 1 8 1 2
------------------------------------------------------------------------
7 3 107 2 2 45 3 7 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 27 1 2 72 1 2 28 1 2 9 1 2 3 8 1 2 3 3 1 2 4
------------------------------------------------------------------------
10 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
10 5 4 2 1 9
o6 = (map(R,R,{--x + -x + x , x , -x + -x + x , -x + -x + x , x }),
7 1 2 2 5 1 9 1 5 2 4 6 1 4 2 3 2
------------------------------------------------------------------------
10 2 5 3 1000 3 750 2 2 300 2
ideal (--x + -x x + x x - x , ----x x + ---x x + ---x x x +
7 1 2 1 2 1 5 2 343 1 2 49 1 2 49 1 2 5
------------------------------------------------------------------------
375 3 150 2 30 2 125 4 75 3 15 2 2 3
---x x + ---x x x + --x x x + ---x + --x x + --x x + x x ), {x ,
14 1 2 7 1 2 5 7 1 2 5 8 2 4 2 5 2 2 5 2 5 5
------------------------------------------------------------------------
x , x })
4 3
o6 : Sequence
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i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 1120x_1x_2x_5^6-60000x_2^9x_5-109375x_2^9+12000x_2^8x_5^2+43750x
{-9} | 122500x_1x_2^2x_5^3-13440x_1x_2x_5^5+49000x_1x_2x_5^4+720000x_2^
{-9} | 18757812500x_1x_2^3+2058000000x_1x_2^2x_5^2+15006250000x_1x_2^2x
{-3} | 20x_1^2+35x_1x_2+14x_1x_5-14x_2^3
------------------------------------------------------------------------
_2^8x_5-1600x_2^7x_5^3-17500x_2^7x_5^2+7000x_2^6x_5^3-2800x_
9-144000x_2^8x_5-175000x_2^8+19200x_2^7x_5^2+140000x_2^7x_5-
_5+41287680x_1x_2x_5^5-75264000x_1x_2x_5^4+548800000x_1x_2x_
------------------------------------------------------------------------
2^5x_5^4+1120x_2^4x_5^5+1960x_2^2x_5^6+784x_2x_5^7
84000x_2^6x_5^2+33600x_2^5x_5^3-13440x_2^4x_5^4+49000x_2^4x_5^3+214375x_
5^3+3001250000x_1x_2x_5^2-2211840000x_2^9+442368000x_2^8x_5+806400000x_2
------------------------------------------------------------------------
2^3x_5^3-23520x_2^2x_5^5+171500x_2^2x_5^4-9408x_2x_5^6+34300x_2x_5^5
^8-58982400x_2^7x_5^2-537600000x_2^7x_5+392000000x_2^7+258048000x_2^6x_5
------------------------------------------------------------------------
^2-470400000x_2^6x_5-1715000000x_2^6-103219200x_2^5x_5^3+188160000x_2^5x
------------------------------------------------------------------------
_5^2+686000000x_2^5x_5+7503125000x_2^5+41287680x_2^4x_5^4-75264000x_2^4x
------------------------------------------------------------------------
_5^3+548800000x_2^4x_5^2+3001250000x_2^4x_5+32826171875x_2^4+3601500000x
------------------------------------------------------------------------
_2^3x_5^2+39391406250x_2^3x_5+72253440x_2^2x_5^5-131712000x_2^2x_5^4+
------------------------------------------------------------------------
2401000000x_2^2x_5^3+15756562500x_2^2x_5^2+28901376x_2x_5^6-52684800x_2x
------------------------------------------------------------------------
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_5^5+384160000x_2x_5^4+2100875000x_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 4 2 2 2
o13 = (map(R,R,{5x + -x + x , x , -x + -x + x , x }), ideal (6x + -x x
1 3 2 4 1 5 1 5 2 3 2 1 3 1 2
-----------------------------------------------------------------------
3 38 2 2 4 3 2 2 2 4 2
+ x x + 1, 4x x + --x x + --x x + 5x x x + -x x x + -x x x +
1 4 1 2 15 1 2 15 1 2 1 2 3 3 1 2 3 5 1 2 4
-----------------------------------------------------------------------
2 2
-x x x + x x x x + 1), {x , x })
5 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
1 5 5 8 2
o16 = (map(R,R,{-x + 3x + x , x , -x + -x + x , x }), ideal (-x + 3x x
7 1 2 4 1 3 1 9 2 3 2 7 1 1 2
-----------------------------------------------------------------------
5 3 320 2 2 5 3 1 2 2 5 2
+ x x + 1, --x x + ---x x + -x x + -x x x + 3x x x + -x x x +
1 4 21 1 2 63 1 2 3 1 2 7 1 2 3 1 2 3 3 1 2 4
-----------------------------------------------------------------------
5 2
-x x x + x x x x + 1), {x , x })
9 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 - 2x + x , x , - 2x + x + x , x }), ideal (3x - 2x x
1 2 4 1 1 2 3 2 1 1 2
-----------------------------------------------------------------------
3 2 2 3 2 2 2
+ x x + 1, - 4x x + 6x x - 2x x + 2x x x - 2x x x - 2x x x +
1 4 1 2 1 2 1 2 1 2 3 1 2 3 1 2 4
-----------------------------------------------------------------------
2
x 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.