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 8 2
o3 = (map(R,R,{5x + 2x + x , x , -x + -x + x , x }), ideal (6x + 2x x +
1 2 4 1 6 1 5 2 3 2 1 1 2
------------------------------------------------------------------------
35 3 31 2 2 16 3 2 2 7 2
x x + 1, --x x + --x x + --x x + 5x x x + 2x x x + -x x x +
1 4 6 1 2 3 1 2 5 1 2 1 2 3 1 2 3 6 1 2 4
------------------------------------------------------------------------
8 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
1 7 4 5
o6 = (map(R,R,{-x + -x + x , x , -x + -x + x , x + 3x + x , x }), ideal
5 1 6 2 5 1 9 1 8 2 4 1 2 3 2
------------------------------------------------------------------------
1 2 7 3 1 3 7 2 2 3 2 49 3 7 2
(-x + -x x + x x - x , ---x x + --x x + --x x x + --x x + -x x x
5 1 6 1 2 1 5 2 125 1 2 50 1 2 25 1 2 5 60 1 2 5 1 2 5
------------------------------------------------------------------------
3 2 343 4 49 3 7 2 2 3
+ -x x x + ---x + --x x + -x x + x x ), {x , x , x })
5 1 2 5 216 2 12 2 5 2 2 5 2 5 5 4 3
o6 : Sequence
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i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 38880x_1x_2x_5^6-63504x_2^9x_5-84035x_2^9+27216x_2^8x_5^2+72030x
{-9} | 360150x_1x_2^2x_5^3-116640x_1x_2x_5^5+308700x_1x_2x_5^4+190512x_
{-9} | 7414975286250x_1x_2^3+2401451388000x_1x_2^2x_5^2+12711386205000x
{-3} | 6x_1^2+35x_1x_2+30x_1x_5-30x_2^3
------------------------------------------------------------------------
_2^8x_5-7776x_2^7x_5^3-61740x_2^7x_5^2+52920x_2^6x_5^3-
2^9-81648x_2^8x_5-72030x_2^8+23328x_2^7x_5^2+123480x_2^
_1x_2^2x_5+195910410240x_1x_2x_5^5-259248729600x_1x_2x_
------------------------------------------------------------------------
45360x_2^5x_5^4+38880x_2^4x_5^5+226800x_2^2x_5^6+194400x_2x_5^7
7x_5-158760x_2^6x_5^2+136080x_2^5x_5^3-116640x_2^4x_5^4+308700x_2^4x_5^3
5^4+1372257936000x_1x_2x_5^3+5447736945000x_1x_2x_5^2-319987003392x_2^9+
------------------------------------------------------------------------
+2100875x_2^3x_5^3-680400x_2^2x_5^5+3601500x_2^2x_5^4-583200x
137137287168x_2^8x_5+181474110720x_2^8-39182082048x_2^7x_5^2-
------------------------------------------------------------------------
_2x_5^6+1543500x_2x_5^5
259248729600x_2^7x_5+137225793600x_2^7+266655836160x_2^6x_5^2-
------------------------------------------------------------------------
352866326400x_2^6x_5-933897762000x_2^6-228562145280x_2^5x_5^3+
------------------------------------------------------------------------
302456851200x_2^5x_5^2+800483796000x_2^5x_5+6355693102500x_2^5+
------------------------------------------------------------------------
195910410240x_2^4x_5^4-259248729600x_2^4x_5^3+1372257936000x_2^4x_5^2+
------------------------------------------------------------------------
5447736945000x_2^4x_5+43254022503125x_2^4+14008466430000x_2^3x_5^2+
------------------------------------------------------------------------
111224629293750x_2^3x_5+1142810726400x_2^2x_5^5-1512284256000x_2^2x_5^4+
------------------------------------------------------------------------
20012094900000x_2^2x_5^3+95335396537500x_2^2x_5^2+979552051200x_2x_5^6-
------------------------------------------------------------------------
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1296243648000x_2x_5^5+6861289680000x_2x_5^4+27238684725000x_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
9 2 7 3 17 2
o13 = (map(R,R,{-x + -x + x , x , -x + --x + x , x }), ideal (--x +
8 1 9 2 4 1 8 1 10 2 3 2 8 1
-----------------------------------------------------------------------
2 63 3 383 2 2 1 3 9 2 2 2
-x x + x x + 1, --x x + ---x x + --x x + -x x x + -x x x +
9 1 2 1 4 64 1 2 720 1 2 15 1 2 8 1 2 3 9 1 2 3
-----------------------------------------------------------------------
7 2 3 2
-x x x + --x x x + x x x x + 1), {x , x })
8 1 2 4 10 1 2 4 1 2 3 4 4 3
o13 : Sequence
|
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
6 4 7 7 13 2 4
o16 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (--x + -x x
7 1 3 2 4 1 8 1 6 2 3 2 7 1 3 1 2
-----------------------------------------------------------------------
3 3 13 2 2 14 3 6 2 4 2 7 2
+ x x + 1, -x x + --x x + --x x + -x x x + -x x x + -x x x +
1 4 4 1 2 6 1 2 9 1 2 7 1 2 3 3 1 2 3 8 1 2 4
-----------------------------------------------------------------------
7 2
-x x x + x x x x + 1), {x , x })
6 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 + x , x }), ideal (x + 2x x + x x +
2 4 1 1 2 3 2 1 1 2 1 4
-----------------------------------------------------------------------
2 2 3 2 2 2
1, 2x x - 2x x + 2x x x + x x x - x x x + x x x x + 1), {x , x })
1 2 1 2 1 2 3 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.