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 5 1 4 11 2 5
o3 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (--x + -x x
7 1 8 2 4 1 5 1 5 2 3 2 7 1 8 1 2
------------------------------------------------------------------------
4 3 163 2 2 1 3 4 2 5 2 1 2
+ x x + 1, --x x + ---x x + -x x + -x x x + -x x x + -x x x +
1 4 35 1 2 280 1 2 2 1 2 7 1 2 3 8 1 2 3 5 1 2 4
------------------------------------------------------------------------
4 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
2 1 9 7
o6 = (map(R,R,{-x + 4x + x , x , -x + x + x , -x + -x + x , x }), ideal
3 1 2 5 1 3 1 2 4 4 1 6 2 3 2
------------------------------------------------------------------------
2 2 3 8 3 16 2 2 4 2 3 2
(-x + 4x x + x x - x , --x x + --x x + -x x x + 32x x + 16x x x
3 1 1 2 1 5 2 27 1 2 3 1 2 3 1 2 5 1 2 1 2 5
------------------------------------------------------------------------
2 4 3 2 2 3
+ 2x x x + 64x + 48x x + 12x x + x x ), {x , x , x })
1 2 5 2 2 5 2 5 2 5 5 4 3
o6 : Sequence
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i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 6x_1x_2x_5^6-384x_2^9x_5-6144x_2^9+48x_2^8x_5^2+1536x_2^8x_5-4x_
{-9} | 768x_1x_2^2x_5^3-6x_1x_2x_5^5+192x_1x_2x_5^4+384x_2^9-48x_2^8x_5
{-9} | 4718592x_1x_2^3+36864x_1x_2^2x_5^2+2359296x_1x_2^2x_5+6x_1x_2x_5
{-3} | 2x_1^2+12x_1x_2+3x_1x_5-3x_2^3
------------------------------------------------------------------------
2^7x_5^3-384x_2^7x_5^2+96x_2^6x_5^3-24x_2^5x_5^4+6x_2^4x_5^5+36x_2^2x_5
-512x_2^8+4x_2^7x_5^2+256x_2^7x_5-96x_2^6x_5^2+24x_2^5x_5^3-6x_2^4x_5^4
^5-96x_1x_2x_5^4+6144x_1x_2x_5^3+294912x_1x_2x_5^2-384x_2^9+48x_2^8x_5+
------------------------------------------------------------------------
^6+9x_2x_5^7
+192x_2^4x_5^3+4608x_2^3x_5^3-36x_2^2x_5^5+2304x_2^2x_5^4-9x_2x_5^6+
768x_2^8-4x_2^7x_5^2-320x_2^7x_5+2048x_2^7+96x_2^6x_5^2-1536x_2^6x_5
------------------------------------------------------------------------
288x_2x_5^5
-49152x_2^6-24x_2^5x_5^3+384x_2^5x_5^2+12288x_2^5x_5+1179648x_2^5+6x_2^
------------------------------------------------------------------------
4x_5^4-96x_2^4x_5^3+6144x_2^4x_5^2+294912x_2^4x_5+28311552x_2^4+221184x_
------------------------------------------------------------------------
2^3x_5^2+21233664x_2^3x_5+36x_2^2x_5^5-576x_2^2x_5^4+92160x_2^2x_5^3+
------------------------------------------------------------------------
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5308416x_2^2x_5^2+9x_2x_5^6-144x_2x_5^5+9216x_2x_5^4+442368x_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
2 7 10 5 2 7
o13 = (map(R,R,{-x + -x + x , x , 3x + --x + x , x }), ideal (-x + -x x
3 1 6 2 4 1 1 3 2 3 2 3 1 6 1 2
-----------------------------------------------------------------------
3 103 2 2 35 3 2 2 7 2 2
+ x x + 1, 2x x + ---x x + --x x + -x x x + -x x x + 3x x x +
1 4 1 2 18 1 2 9 1 2 3 1 2 3 6 1 2 3 1 2 4
-----------------------------------------------------------------------
10 2
--x x x + x x x x + 1), {x , x })
3 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
4 3 1 6 11 2 3
o16 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (--x + -x x
7 1 2 2 4 1 2 1 7 2 3 2 7 1 2 1 2
-----------------------------------------------------------------------
2 3 243 2 2 9 3 4 2 3 2 1 2
+ x x + 1, -x x + ---x x + -x x + -x x x + -x x x + -x x x +
1 4 7 1 2 196 1 2 7 1 2 7 1 2 3 2 1 2 3 2 1 2 4
-----------------------------------------------------------------------
6 2
-x x x + x x x x + 1), {x , x })
7 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
--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
--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: 3
--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: 4
--trying with basis element limit: 5
--trying with basis element limit: 20
2
o19 = (map(R,R,{x + 4x + x , x , x - 8x + x , x }), ideal (2x + 4x x +
1 2 4 1 1 2 3 2 1 1 2
-----------------------------------------------------------------------
3 2 2 3 2 2 2 2
x x + 1, x x - 4x x - 32x x + x x x + 4x x x + x x x - 8x 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.