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 1 5 2 1
o3 = (map(R,R,{2x + -x + x , x , -x + -x + x , x }), ideal (3x + -x x +
1 5 2 4 1 2 1 2 2 3 2 1 5 1 2
------------------------------------------------------------------------
3 51 2 2 1 3 2 1 2 1 2 5 2
x x + 1, x x + --x x + -x x + 2x x x + -x x x + -x x x + -x x x
1 4 1 2 10 1 2 2 1 2 1 2 3 5 1 2 3 2 1 2 4 2 1 2 4
------------------------------------------------------------------------
+ x x x x + 1), {x , x })
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
9 8
o6 = (map(R,R,{-x + 2x + x , x , x + 6x + x , 2x + -x + x , x }), ideal
7 1 2 5 1 1 2 4 1 9 2 3 2
------------------------------------------------------------------------
9 2 3 729 3 486 2 2 243 2 108 3
(-x + 2x 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
------------------------------------------------------------------------
108 2 27 2 4 3 2 2 3
---x x x + --x x x + 8x + 12x x + 6x x + x x ), {x , x , x })
7 1 2 5 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} | 63x_1x_2x_5^6-1944x_2^9x_5-2016x_2^9+486x_2^8x_5^2+1008x_2
{-9} | 7056x_1x_2^2x_5^3-1701x_1x_2x_5^5+3528x_1x_2x_5^4+52488x_2
{-9} | 22127616x_1x_2^3+5334336x_1x_2^2x_5^2+22127616x_1x_2^2x_5+
{-3} | 9x_1^2+14x_1x_2+7x_1x_5-7x_2^3
------------------------------------------------------------------------
^8x_5-81x_2^7x_5^3-504x_2^7x_5^2+252x_2^6x_5^3-126x_2^5x_5^4+63x_2^4x_
^9-13122x_2^8x_5-9072x_2^8+2187x_2^7x_5^2+9072x_2^7x_5-6804x_2^6x_5^2+
413343x_1x_2x_5^5-428652x_1x_2x_5^4+1778112x_1x_2x_5^3+5531904x_1x_2x_
------------------------------------------------------------------------
5^5+98x_2^2x_5^6+49x_2x_5^7
3402x_2^5x_5^3-1701x_2^4x_5^4+3528x_2^4x_5^3+10976x_2^3x_5^3-2646x_2^2x_
5^2-12754584x_2^9+3188646x_2^8x_5+3306744x_2^8-531441x_2^7x_5^2-2755620x
------------------------------------------------------------------------
5^5+10976x_2^2x_5^4-1323x_2x_5^6+2744x_2x_5^5
_2^7x_5+1143072x_2^7+1653372x_2^6x_5^2-1714608x_2^6x_5-3556224x_2^6-
------------------------------------------------------------------------
826686x_2^5x_5^3+857304x_2^5x_5^2+1778112x_2^5x_5+11063808x_2^5+413343x_
------------------------------------------------------------------------
2^4x_5^4-428652x_2^4x_5^3+1778112x_2^4x_5^2+5531904x_2^4x_5+34420736x_2^
------------------------------------------------------------------------
4+8297856x_2^3x_5^2+51631104x_2^3x_5+642978x_2^2x_5^5-666792x_2^2x_5^4+
------------------------------------------------------------------------
6914880x_2^2x_5^3+25815552x_2^2x_5^2+321489x_2x_5^6-333396x_2x_5^5+
------------------------------------------------------------------------
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1382976x_2x_5^4+4302592x_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
5 7 2 9 8 2 7
o13 = (map(R,R,{-x + -x + x , x , -x + --x + x , x }), ideal (-x + -x x
3 1 9 2 4 1 7 1 10 2 3 2 3 1 9 1 2
-----------------------------------------------------------------------
10 3 31 2 2 7 3 5 2 7 2 2 2
+ x x + 1, --x x + --x x + --x x + -x x x + -x x x + -x x x +
1 4 21 1 2 18 1 2 10 1 2 3 1 2 3 9 1 2 3 7 1 2 4
-----------------------------------------------------------------------
9 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
9 5 8 1 14 2 5
o16 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (--x + -x x
5 1 4 2 4 1 7 1 2 2 3 2 5 1 4 1 2
-----------------------------------------------------------------------
72 3 163 2 2 5 3 9 2 5 2 8 2
+ x x + 1, --x x + ---x x + -x x + -x x x + -x x x + -x x x +
1 4 35 1 2 70 1 2 8 1 2 5 1 2 3 4 1 2 3 7 1 2 4
-----------------------------------------------------------------------
1 2
-x x x + x x x x + 1), {x , x })
2 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
--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,{4x - 2x + x , x , - x + x , x }), ideal (5x - 2x x +
1 2 4 1 2 3 2 1 1 2
-----------------------------------------------------------------------
2 2 3 2 2 2
x x + 1, - 4x x + 2x x + 4x x x - 2x x x - x x x + x x x x + 1),
1 4 1 2 1 2 1 2 3 1 2 3 1 2 4 1 2 3 4
-----------------------------------------------------------------------
{x , x })
4 3
o19 : Sequence
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This symbol is provided by the package NoetherNormalization.