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
2 3 5 2
o3 = (map(R,R,{-x + 2x + x , x , 3x + -x + x , x }), ideal (-x + 2x x +
3 1 2 4 1 1 7 2 3 2 3 1 1 2
------------------------------------------------------------------------
3 44 2 2 6 3 2 2 2 2 3 2
x x + 1, 2x x + --x x + -x x + -x x x + 2x x x + 3x x x + -x x x
1 4 1 2 7 1 2 7 1 2 3 1 2 3 1 2 3 1 2 4 7 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
10 1 3 3
o6 = (map(R,R,{--x + 3x + x , x , -x + 9x + x , -x + -x + x , x }),
7 1 2 5 1 2 1 2 4 2 1 4 2 3 2
------------------------------------------------------------------------
10 2 3 1000 3 900 2 2 300 2
ideal (--x + 3x 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
------------------------------------------------------------------------
270 3 180 2 30 2 4 3 2 2 3
---x x + ---x x x + --x x x + 27x + 27x x + 9x x + x x ), {x , x ,
7 1 2 7 1 2 5 7 1 2 5 2 2 5 2 5 2 5 5 4
------------------------------------------------------------------------
x })
3
o6 : Sequence
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i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 70x_1x_2x_5^6-5400x_2^9x_5-17010x_2^9+900x_2^8x_5^2+5670x_2^
{-9} | 13230x_1x_2^2x_5^3-700x_1x_2x_5^5+4410x_1x_2x_5^4+54000x_2^9
{-9} | 472588830x_1x_2^3+25004700x_1x_2^2x_5^2+315059220x_1x_2^2x_5
{-3} | 10x_1^2+21x_1x_2+7x_1x_5-7x_2^3
------------------------------------------------------------------------
8x_5-100x_2^7x_5^3-1890x_2^7x_5^2+630x_2^6x_5^3-210x_2^5x_5^4+70x_2^
-9000x_2^8x_5-18900x_2^8+1000x_2^7x_5^2+12600x_2^7x_5-6300x_2^6x_5^2
+140000x_1x_2x_5^5-441000x_1x_2x_5^4+5556600x_1x_2x_5^3+52509870x_1x
------------------------------------------------------------------------
4x_5^5+147x_2^2x_5^6+49x_2x_5^7
+2100x_2^5x_5^3-700x_2^4x_5^4+4410x_2^4x_5^3+27783x_2^3x_5^3-1470x_2
_2x_5^2-10800000x_2^9+1800000x_2^8x_5+5670000x_2^8-200000x_2^7x_5^2-
------------------------------------------------------------------------
^2x_5^5+18522x_2^2x_5^4-490x_2x_5^6+3087x_2x_5^5
3150000x_2^7x_5+3969000x_2^7+1260000x_2^6x_5^2-3969000x_2^6x_5-25004700x
------------------------------------------------------------------------
_2^6-420000x_2^5x_5^3+1323000x_2^5x_5^2+8334900x_2^5x_5+157529610x_2^5+
------------------------------------------------------------------------
140000x_2^4x_5^4-441000x_2^4x_5^3+5556600x_2^4x_5^2+52509870x_2^4x_5+
------------------------------------------------------------------------
992436543x_2^4+52509870x_2^3x_5^2+992436543x_2^3x_5+294000x_2^2x_5^5-
------------------------------------------------------------------------
926100x_2^2x_5^4+29172150x_2^2x_5^3+330812181x_2^2x_5^2+98000x_2x_5^6-
------------------------------------------------------------------------
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308700x_2x_5^5+3889620x_2x_5^4+36756909x_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
1 1 4 7 2
o13 = (map(R,R,{-x + x + x , x , -x + -x + x , x }), ideal (-x + x x +
6 1 2 4 1 2 1 7 2 3 2 6 1 1 2
-----------------------------------------------------------------------
1 3 25 2 2 4 3 1 2 2 1 2
x x + 1, --x x + --x x + -x x + -x x x + x x x + -x x x +
1 4 12 1 2 42 1 2 7 1 2 6 1 2 3 1 2 3 2 1 2 4
-----------------------------------------------------------------------
4 2
-x x x + x x x x + 1), {x , x })
7 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 1 9 7 16 2
o16 = (map(R,R,{-x + -x + x , x , -x + --x + x , x }), ideal (--x +
7 1 3 2 4 1 5 1 10 2 3 2 7 1
-----------------------------------------------------------------------
1 81 3 3 2 2 7 3 9 2 1 2 9 2
-x x + x x + 1, --x x + -x x + --x x + -x x x + -x x x + -x x x
3 1 2 1 4 35 1 2 2 1 2 30 1 2 7 1 2 3 3 1 2 3 5 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
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
--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,{2x - 2x + x , x , 2x - 5x + 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 - 14x x + 10x 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
5x 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.