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 10 2 1 8 2
o3 = (map(R,R,{-x + --x + x , x , -x + --x + x , x }), ideal (-x +
7 1 7 2 4 1 3 1 10 2 3 2 7 1
------------------------------------------------------------------------
10 2 3 29 2 2 1 3 1 2 10 2
--x x + x x + 1, --x x + --x x + -x x + -x x x + --x x x +
7 1 2 1 4 21 1 2 30 1 2 7 1 2 7 1 2 3 7 1 2 3
------------------------------------------------------------------------
2 2 1 2
-x x x + --x x x + x x x x + 1), {x , x })
3 1 2 4 10 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
3 9 3 1 1
o6 = (map(R,R,{-x + --x + x , x , -x + x + x , -x + -x + x , x }),
2 1 10 2 5 1 8 1 2 4 5 1 2 2 3 2
------------------------------------------------------------------------
3 2 9 3 27 3 243 2 2 27 2 729 3
ideal (-x + --x x + x x - x , --x x + ---x x + --x x x + ---x x +
2 1 10 1 2 1 5 2 8 1 2 40 1 2 4 1 2 5 200 1 2
------------------------------------------------------------------------
81 2 9 2 729 4 243 3 27 2 2 3
--x x x + -x x x + ----x + ---x x + --x x + x x ), {x , x , x })
10 1 2 5 2 1 2 5 1000 2 100 2 5 10 2 5 2 5 5 4 3
o6 : Sequence
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i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 300000x_1x_2x_5^6-2187000x_2^9x_5-177147x_2^9+1215000x_2^8x_5^2+
{-9} | 21870x_1x_2^2x_5^3-150000x_1x_2x_5^5+24300x_1x_2x_5^4+1093500x_2
{-9} | 1937102445x_1x_2^3+13286025000x_1x_2^2x_5^2+4304672100x_1x_2^2x_
{-3} | 15x_1^2+9x_1x_2+10x_1x_5-10x_2^3
------------------------------------------------------------------------
196830x_2^8x_5-450000x_2^7x_5^3-218700x_2^7x_5
^9-607500x_2^8x_5-32805x_2^8+225000x_2^7x_5^2+
5+375000000000x_1x_2x_5^5-30375000000x_1x_2x_5
------------------------------------------------------------------------
^2+243000x_2^6x_5^3-270000x_2^5x_5^4+300000x_2^4x_5^5+180000x_2^
72900x_2^7x_5-121500x_2^6x_5^2+135000x_2^5x_5^3-150000x_2^4x_5^4
^4+9841500000x_1x_2x_5^3+2391484500x_1x_2x_5^2-2733750000000x_2^
------------------------------------------------------------------------
2x_5^6+200000x_2x_5^7
+24300x_2^4x_5^3+13122x_2^3x_5^3-90000x_2^2x_5^5+29160x_2^2x_5^4-
9+1518750000000x_2^8x_5+123018750000x_2^8-562500000000x_2^7x_5^2-
------------------------------------------------------------------------
100000x_2x_5^6+16200x_2x_5^5
227812500000x_2^7x_5+7381125000x_2^7+303750000000x_2^6x_5^2-24603750000x
------------------------------------------------------------------------
_2^6x_5-3985807500x_2^6-337500000000x_2^5x_5^3+27337500000x_2^5x_5^2+
------------------------------------------------------------------------
4428675000x_2^5x_5+2152336050x_2^5+375000000000x_2^4x_5^4-30375000000x_2
------------------------------------------------------------------------
^4x_5^3+9841500000x_2^4x_5^2+2391484500x_2^4x_5+1162261467x_2^4+
------------------------------------------------------------------------
7971615000x_2^3x_5^2+3874204890x_2^3x_5+225000000000x_2^2x_5^5-
------------------------------------------------------------------------
18225000000x_2^2x_5^4+14762250000x_2^2x_5^3+4304672100x_2^2x_5^2+
------------------------------------------------------------------------
250000000000x_2x_5^6-20250000000x_2x_5^5+6561000000x_2x_5^4+1594323000x_
------------------------------------------------------------------------
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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 6 7 2 1
o13 = (map(R,R,{x + -x + x , x , -x + -x + x , x }), ideal (2x + -x x +
1 9 2 4 1 5 1 9 2 3 2 1 9 1 2
-----------------------------------------------------------------------
6 3 41 2 2 7 3 2 1 2 6 2
x x + 1, -x x + --x x + --x x + x x x + -x x x + -x x x +
1 4 5 1 2 45 1 2 81 1 2 1 2 3 9 1 2 3 5 1 2 4
-----------------------------------------------------------------------
7 2
-x x x + x x x x + 1), {x , x })
9 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
2 4 4 2 2
o16 = (map(R,R,{x + -x + x , x , -x + -x + x , x }), ideal (2x + -x x +
1 3 2 4 1 9 1 9 2 3 2 1 3 1 2
-----------------------------------------------------------------------
4 3 20 2 2 8 3 2 2 2 4 2
x x + 1, -x x + --x x + --x x + x x x + -x x x + -x x x +
1 4 9 1 2 27 1 2 27 1 2 1 2 3 3 1 2 3 9 1 2 4
-----------------------------------------------------------------------
4 2
-x x x + x x x x + 1), {x , x })
9 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 , x }), ideal (3x + x x +
1 2 4 1 1 2 3 2 1 1 2
-----------------------------------------------------------------------
3 2 2 3 2 2 2 2
x x + 1, - 2x x + x x + x x + 2x x x + x x x - x x x + x x x +
1 4 1 2 1 2 1 2 1 2 3 1 2 3 1 2 4 1 2 4
-----------------------------------------------------------------------
x x x x + 1), {x , x })
1 2 3 4 4 3
o19 : Sequence
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This symbol is provided by the package NoetherNormalization.