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 3 8 2
o3 = (map(R,R,{-x + x + x , x , -x + 2x + x , x }), ideal (-x + x x +
7 1 2 4 1 5 1 2 3 2 7 1 1 2
------------------------------------------------------------------------
3 3 31 2 2 3 1 2 2 3 2 2
x x + 1, --x x + --x x + 2x x + -x x x + x x x + -x x x + 2x x x
1 4 35 1 2 35 1 2 1 2 7 1 2 3 1 2 3 5 1 2 4 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
4 7 5 1
o6 = (map(R,R,{-x + -x + x , x , x + x + x , -x + -x + x , x }), ideal
7 1 9 2 5 1 1 2 4 2 1 2 2 3 2
------------------------------------------------------------------------
4 2 7 3 64 3 16 2 2 48 2 28 3 8 2
(-x + -x x + x x - x , ---x x + --x x + --x x x + --x x + -x x x
7 1 9 1 2 1 5 2 343 1 2 21 1 2 49 1 2 5 27 1 2 3 1 2 5
------------------------------------------------------------------------
12 2 343 4 49 3 7 2 2 3
+ --x x x + ---x + --x x + -x x + x x ), {x , x , x })
7 1 2 5 729 2 27 2 5 3 2 5 2 5 5 4 3
o6 : Sequence
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i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 1653372x_1x_2x_5^
{-9} | 4235364x_1x_2^2x_
{-9} | 170912214357948x_
{-3} | 36x_1^2+49x_1x_2+
------------------------------------------------------------------------
6-3429216x_2^9x_5-470596x_2^9+2204496x_2^8x_5^2+605052x_2
5^3-19840464x_1x_2x_5^5+5445468x_1x_2x_5^4+41150592x_2^9-
1x_2^3+800634286953648x_1x_2^2x_5^2+439488551206152x_1x_2
63x_1x_5-63x_2^3
------------------------------------------------------------------------
^8x_5-944784x_2^7x_5^3-777924x_2^7x_5^2+1000188x_2^6x_5^3-
26453952x_2^8x_5-2420208x_2^8+11337408x_2^7x_5^2+6223392x_
^2x_5+9110047128731136x_1x_2x_5^5-1250184222455616x_1x_2x_
------------------------------------------------------------------------
1285956x_2^5x_5^4+1653372x_2^4x_5^5+2250423x_2^2x_5^6+
2^7x_5-12002256x_2^6x_5^2+15431472x_2^5x_5^3-19840464x
5^4+686257960245984x_1x_2x_5^3+282528354346812x_1x_2x_
------------------------------------------------------------------------
2893401x_2x_5^7
_2^4x_5^4+5445468x_2^4x_5^3+5764801x_2^3x_5^3-
5^2-18894912563294208x_2^9+12146729504974848x_
------------------------------------------------------------------------
27005076x_2^2x_5^5+14823774x_2^2x_5^4-34720812x_2x_5^6+
2^8x_5+1666912296607488x_2^8-5205741216417792x_2^7x_5^2
------------------------------------------------------------------------
9529569x_2x_5^5
-3571954921301760x_2^7x_5+196073702927424x_2^7+5511016164294144x_2^6x_5^
------------------------------------------------------------------------
2-756284282720064x_2^6x_5-207571852173168x_2^6-7085592211235328x_2^5x_5^
------------------------------------------------------------------------
3+972365506354368x_2^5x_5^2+266878095651216x_2^5x_5+219744275603076x_2^5
------------------------------------------------------------------------
+9110047128731136x_2^4x_5^4-1250184222455616x_2^4x_5^3+686257960245984x_
------------------------------------------------------------------------
2^4x_5^2+282528354346812x_2^4x_5+232630513987207x_2^4+1089752223909132x_
------------------------------------------------------------------------
2^3x_5^2+897289125379227x_2^3x_5+12399786369661824x_2^2x_5^5-
------------------------------------------------------------------------
1701639636120144x_2^2x_5^4+2335183336948140x_2^2x_5^3+1153657446916149x_
------------------------------------------------------------------------
2^2x_5^2+15942582475279488x_2x_5^6-2187822389297328x_2x_5^5+
------------------------------------------------------------------------
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1200951430430472x_2x_5^4+494424620106921x_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
1 1 5 2 1
o13 = (map(R,R,{4x + -x + x , x , -x + -x + x , x }), ideal (5x + -x x
1 2 2 4 1 6 1 4 2 3 2 1 2 1 2
-----------------------------------------------------------------------
2 3 61 2 2 5 3 2 1 2 1 2
+ x x + 1, -x x + --x x + -x x + 4x x x + -x x x + -x x x +
1 4 3 1 2 12 1 2 8 1 2 1 2 3 2 1 2 3 6 1 2 4
-----------------------------------------------------------------------
5 2
-x x x + x x x x + 1), {x , x })
4 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
9 3 16 2 3
o16 = (map(R,R,{-x + -x + x , x , 8x + 3x + x , x }), ideal (--x + -x x
7 1 2 2 4 1 1 2 3 2 7 1 2 1 2
-----------------------------------------------------------------------
72 3 111 2 2 9 3 9 2 3 2 2
+ x x + 1, --x x + ---x x + -x x + -x x x + -x x x + 8x x x +
1 4 7 1 2 7 1 2 2 1 2 7 1 2 3 2 1 2 3 1 2 4
-----------------------------------------------------------------------
2
3x x x + x x x x + 1), {x , x })
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];
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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.