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
1 5 1 1 11 2 5
o3 = (map(R,R,{--x + -x + x , x , -x + -x + x , x }), ideal (--x + -x x
10 1 9 2 4 1 5 1 5 2 3 2 10 1 9 1 2
------------------------------------------------------------------------
1 3 59 2 2 1 3 1 2 5 2 1 2
+ x x + 1, --x x + ---x x + -x x + --x x x + -x x x + -x x x +
1 4 50 1 2 450 1 2 9 1 2 10 1 2 3 9 1 2 3 5 1 2 4
------------------------------------------------------------------------
1 2
-x x x + x x x x + 1), {x , x })
5 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
5 7 2
o6 = (map(R,R,{5x + -x + x , x , -x + x + x , 4x + -x + x , x }), ideal
1 4 2 5 1 9 1 2 4 1 9 2 3 2
------------------------------------------------------------------------
2 5 3 3 375 2 2 2 375 3
(5x + -x x + x x - x , 125x x + ---x x + 75x x x + ---x x +
1 4 1 2 1 5 2 1 2 4 1 2 1 2 5 16 1 2
------------------------------------------------------------------------
75 2 2 125 4 75 3 15 2 2 3
--x x x + 15x x x + ---x + --x x + --x x + x x ), {x , x , x })
2 1 2 5 1 2 5 64 2 16 2 5 4 2 5 2 5 5 4 3
o6 : Sequence
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i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 5120x_1x_2x_5^6-240000x_2^9x_5-15625x_2^9+96000x_2^8x_5^2+
{-9} | 2500x_1x_2^2x_5^3-15360x_1x_2x_5^5+2000x_1x_2x_5^4+720000x
{-9} | 7812500x_1x_2^3+48000000x_1x_2^2x_5^2+12500000x_1x_2^2x_5+
{-3} | 20x_1^2+5x_1x_2+4x_1x_5-4x_2^3
------------------------------------------------------------------------
12500x_2^8x_5-25600x_2^7x_5^3-10000x_2^7x_5^2+8000x_2^6x_5^3-
_2^9-288000x_2^8x_5-12500x_2^8+76800x_2^7x_5^2+20000x_2^7x_5-
1509949440x_1x_2x_5^5-98304000x_1x_2x_5^4+25600000x_1x_2x_5^3
------------------------------------------------------------------------
6400x_2^5x_5^4+5120x_2^4x_5^5+1280x_2^2x_5^6+1024x_2x_5^7
24000x_2^6x_5^2+19200x_2^5x_5^3-15360x_2^4x_5^4+2000x_2^4x_5^3+625x_2^3x
+5000000x_1x_2x_5^2-70778880000x_2^9+28311552000x_2^8x_5+1843200000x_2^8
------------------------------------------------------------------------
_5^3-3840x_2^2x_5^5+1000x_2^2x_5^4-3072x_2x_5^6+400x_2x_5^5
-7549747200x_2^7x_5^2-2457600000x_2^7x_5+64000000x_2^7+2359296000x_2^6x_
------------------------------------------------------------------------
5^2-153600000x_2^6x_5-20000000x_2^6-1887436800x_2^5x_5^3+122880000x_2^5x
------------------------------------------------------------------------
_5^2+16000000x_2^5x_5+6250000x_2^5+1509949440x_2^4x_5^4-98304000x_2^4x_5
------------------------------------------------------------------------
^3+25600000x_2^4x_5^2+5000000x_2^4x_5+1953125x_2^4+12000000x_2^3x_5^2+
------------------------------------------------------------------------
4687500x_2^3x_5+377487360x_2^2x_5^5-24576000x_2^2x_5^4+16000000x_2^2x_5^
------------------------------------------------------------------------
3+3750000x_2^2x_5^2+301989888x_2x_5^6-19660800x_2x_5^5+5120000x_2x_5^4+
------------------------------------------------------------------------
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1000000x_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
8 10 7 9 15 2
o13 = (map(R,R,{-x + --x + x , x , -x + -x + x , x }), ideal (--x +
7 1 3 2 4 1 2 1 5 2 3 2 7 1
-----------------------------------------------------------------------
10 3 1441 2 2 3 8 2 10 2
--x x + x x + 1, 4x x + ----x x + 6x x + -x x x + --x x x +
3 1 2 1 4 1 2 105 1 2 1 2 7 1 2 3 3 1 2 3
-----------------------------------------------------------------------
7 2 9 2
-x x x + -x x x + x x x x + 1), {x , x })
2 1 2 4 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
3 1 10 7 2
o16 = (map(R,R,{-x + x + x , x , -x + --x + x , x }), ideal (-x + x x +
4 1 2 4 1 3 1 7 2 3 2 4 1 1 2
-----------------------------------------------------------------------
1 3 59 2 2 10 3 3 2 2 1 2
x x + 1, -x x + --x x + --x x + -x x x + x x x + -x x x +
1 4 4 1 2 42 1 2 7 1 2 4 1 2 3 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
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 , x , - x + 2x + x , x }), ideal (- x +
1 2 4 1 1 2 3 2 1
-----------------------------------------------------------------------
3 2 2 3 2 2 2
x x + x x + 1, 2x x - 5x x + 2x x - 2x x x + x x x - x x x +
1 2 1 4 1 2 1 2 1 2 1 2 3 1 2 3 1 2 4
-----------------------------------------------------------------------
2
2x 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.