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 3 7 9 2 3
o3 = (map(R,R,{-x + -x + x , x , -x + 7x + x , x }), ideal (-x + -x x +
5 1 7 2 4 1 2 1 2 3 2 5 1 7 1 2
------------------------------------------------------------------------
14 3 71 2 2 3 4 2 3 2 7 2
x x + 1, --x x + --x x + 3x x + -x x x + -x x x + -x x x +
1 4 5 1 2 10 1 2 1 2 5 1 2 3 7 1 2 3 2 1 2 4
------------------------------------------------------------------------
2
7x x x + x x x x + 1), {x , x })
1 2 4 1 2 3 4 4 3
o3 : Sequence
|
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];
|
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 7 9 3 5
o6 = (map(R,R,{--x + -x + x , x , -x + -x + x , 5x + -x + x , x }),
9 1 8 2 5 1 2 1 4 2 4 1 3 2 3 2
------------------------------------------------------------------------
10 2 7 3 1000 3 175 2 2 100 2
ideal (--x + -x x + x x - x , ----x x + ---x x + ---x x x +
9 1 8 1 2 1 5 2 729 1 2 54 1 2 27 1 2 5
------------------------------------------------------------------------
245 3 35 2 10 2 343 4 147 3 21 2 2 3
---x x + --x x x + --x x x + ---x + ---x x + --x x + x x ), {x ,
96 1 2 6 1 2 5 3 1 2 5 512 2 64 2 5 8 2 5 2 5 5
------------------------------------------------------------------------
x , x })
4 3
o6 : Sequence
|
i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 1474560x_1x_2x_5^
{-9} | 5186160x_1x_2^2x_
{-9} | 115317695651760x_
{-3} | 80x_1^2+63x_1x_2+
------------------------------------------------------------------------
6-7526400x_2^9x_5-756315x_2^9+4300800x_2^8x_5^2+864360x_2^
5^3-29491200x_1x_2x_5^5+5927040x_1x_2x_5^4+150528000x_2^9-
1x_2^3+655756325683200x_1x_2^2x_5^2+263583304346880x_1x_2^
72x_1x_5-72x_2^3
------------------------------------------------------------------------
8x_5-1638400x_2^7x_5^3-987840x_2^7x_5^2+1128960x_2^6x_5^3-
86016000x_2^8x_5-5762400x_2^8+32768000x_2^7x_5^2+13171200x
2x_5+12369505812480000x_1x_2x_5^5-1242990379008000x_1x_2x_
------------------------------------------------------------------------
1290240x_2^5x_5^4+1474560x_
_2^7x_5-22579200x_2^6x_5^2+
5^4+499623867187200x_1x_2x_
------------------------------------------------------------------------
2^4x_5^5+1161216x_2^2x_5^6+1327104x_2x_5^7
25804800x_2^5x_5^3-29491200x_2^4x_5^4+5927040x_2^4x_5^3+4084101x_2^3x_5^
5^3+150619031055360x_1x_2x_5^2-63136019251200000x_2^9+36077725286400000x
------------------------------------------------------------------------
3-23224320x_2^2x_5^5+9335088x_2^2x_5^4-26542080x_2x_5^6+
_2^8x_5+3625388605440000x_2^8-13743895347200000x_2^7x_5^
------------------------------------------------------------------------
5334336x_2x_5^5
2-6905502105600000x_2^7x_5+277568815104000x_2^7+9470402887680000x_2^6x_5
------------------------------------------------------------------------
^2-951664508928000x_2^6x_5-191262261657600x_2^6-10823317585920000x_2^5x_
------------------------------------------------------------------------
5^3+1087616581632000x_2^5x_5^2+218585441894400x_2^5x_5+131791652173440x_
------------------------------------------------------------------------
2^5+12369505812480000x_2^4x_5^4-1242990379008000x_2^4x_5^3+
------------------------------------------------------------------------
499623867187200x_2^4x_5^2+150619031055360x_2^4x_5+90812685325761x_2^4+
------------------------------------------------------------------------
516408106475520x_2^3x_5^2+311357778259752x_2^3x_5+9740985827328000x_2^2x
------------------------------------------------------------------------
_5^5-978854923468800x_2^2x_5^4+983634488524800x_2^2x_5^3+
------------------------------------------------------------------------
355837460868288x_2^2x_5^2+11132555231232000x_2x_5^6-1118691341107200x_2x
------------------------------------------------------------------------
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|
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_5^5+449661480468480x_2x_5^4+135557127949824x_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];
|
i9 : I = ideal(a^2*b+a*b^2+1);
o9 : Ideal of R
|
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];
|
i12 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);
o12 : Ideal of R
|
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 7 2
o13 = (map(R,R,{x + 4x + x , x , --x + --x + x , x }), ideal (2x + 4x x
1 2 4 1 10 1 10 2 3 2 1 1 2
-----------------------------------------------------------------------
3 3 19 2 2 14 3 2 2 3 2
+ x x + 1, --x x + --x x + --x x + x x x + 4x x x + --x x x +
1 4 10 1 2 10 1 2 5 1 2 1 2 3 1 2 3 10 1 2 4
-----------------------------------------------------------------------
7 2
--x x x + x x x x + 1), {x , x })
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];
|
i15 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);
o15 : Ideal of R
|
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 8 4 3 6 2 8
o16 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (-x + -x x
5 1 5 2 4 1 5 1 7 2 3 2 5 1 5 1 2
-----------------------------------------------------------------------
4 3 239 2 2 24 3 1 2 8 2 4 2
+ x x + 1, --x x + ---x x + --x x + -x x x + -x x x + -x x x +
1 4 25 1 2 175 1 2 35 1 2 5 1 2 3 5 1 2 3 5 1 2 4
-----------------------------------------------------------------------
3 2
-x x x + x x x x + 1), {x , x })
7 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];
|
i18 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);
o18 : Ideal of R
|
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 , - 4x - 2x + x , x }), ideal (5x +
1 2 4 1 1 2 3 2 1
-----------------------------------------------------------------------
3 2 2 3 2 2
2x x + x x + 1, - 16x x - 16x x - 4x x + 4x x x + 2x x x -
1 2 1 4 1 2 1 2 1 2 1 2 3 1 2 3
-----------------------------------------------------------------------
2 2
4x x x - 2x x x + x x x x + 1), {x , x })
1 2 4 1 2 4 1 2 3 4 4 3
o19 : Sequence
|
This symbol is provided by the package NoetherNormalization.