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
10 5 19 2
o3 = (map(R,R,{--x + 3x + x , x , x + -x + x , x }), ideal (--x + 3x x
9 1 2 4 1 1 6 2 3 2 9 1 1 2
------------------------------------------------------------------------
10 3 106 2 2 5 3 10 2 2 2
+ x x + 1, --x x + ---x x + -x x + --x x x + 3x x x + x x x +
1 4 9 1 2 27 1 2 2 1 2 9 1 2 3 1 2 3 1 2 4
------------------------------------------------------------------------
5 2
-x x x + x x x x + 1), {x , x })
6 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
5 1 10 3 5
o6 = (map(R,R,{-x + -x + x , x , --x + -x + x , 3x + -x + x , x }),
8 1 3 2 5 1 7 1 4 2 4 1 2 2 3 2
------------------------------------------------------------------------
5 2 1 3 125 3 25 2 2 75 2 5 3
ideal (-x + -x x + x x - x , ---x x + --x x + --x x x + --x x +
8 1 3 1 2 1 5 2 512 1 2 64 1 2 64 1 2 5 24 1 2
------------------------------------------------------------------------
5 2 15 2 1 4 1 3 2 2 3
-x x x + --x x x + --x + -x x + x x + x x ), {x , x , x })
4 1 2 5 8 1 2 5 27 2 3 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} | 9720x_1x_2x_5^6-4050x_2^9x_5-40x_2^9+6075x_2^8x_5^2
{-9} | 960x_1x_2^2x_5^3-145800x_1x_2x_5^5+2880x_1x_2x_5^4+
{-9} | 30720x_1x_2^3+4665600x_1x_2^2x_5^2+184320x_1x_2^2x_
{-3} | 15x_1^2+8x_1x_2+24x_1x_5-24x_2^3
------------------------------------------------------------------------
+120x_2^8x_5-6075x_2^7x_5^3-360x_2^7x_5^2+1080x_2^6x_5^3-3240x_2^5x_5^4+
60750x_2^9-91125x_2^8x_5-600x_2^8+91125x_2^7x_5^2+3600x_2^7x_5-16200x_2^
5+23914845000x_1x_2x_5^5-236196000x_1x_2x_5^4+9331200x_1x_2x_5^3+276480x
------------------------------------------------------------------------
9720x_2^4x_5^5+5184x_2^2x_5^6+15552x_2x_5^7
6x_5^2+48600x_2^5x_5^3-145800x_2^4x_5^4+2880x_2^4x_5^3+512x_2^
_1x_2x_5^2-9964518750x_2^9+14946778125x_2^8x_5+147622500x_2^8-
------------------------------------------------------------------------
3x_5^3-77760x_2^2x_5^5+3072x_2^2x_5^4-233280x_2x_5^6+4608x_2x_5^5
14946778125x_2^7x_5^2-738112500x_2^7x_5+2916000x_2^7+2657205000x_2^6x_5^
------------------------------------------------------------------------
2-26244000x_2^6x_5-518400x_2^6-7971615000x_2^5x_5^3+78732000x_2^5x_5^2+
------------------------------------------------------------------------
1555200x_2^5x_5+92160x_2^5+23914845000x_2^4x_5^4-236196000x_2^4x_5^3+
------------------------------------------------------------------------
9331200x_2^4x_5^2+276480x_2^4x_5+16384x_2^4+2488320x_2^3x_5^2+147456x_2^
------------------------------------------------------------------------
3x_5+12754584000x_2^2x_5^5-125971200x_2^2x_5^4+12441600x_2^2x_5^3+
------------------------------------------------------------------------
442368x_2^2x_5^2+38263752000x_2x_5^6-377913600x_2x_5^5+14929920x_2x_5^4+
------------------------------------------------------------------------
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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,{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
3 7 1 2 3
o13 = (map(R,R,{4x + -x + x , x , -x + -x + x , x }), ideal (5x + -x x
1 4 2 4 1 4 1 9 2 3 2 1 4 1 2
-----------------------------------------------------------------------
3 253 2 2 1 3 2 3 2 7 2
+ x x + 1, 7x x + ---x x + --x x + 4x x x + -x x x + -x x x +
1 4 1 2 144 1 2 12 1 2 1 2 3 4 1 2 3 4 1 2 4
-----------------------------------------------------------------------
1 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
4 7 8 4 11 2 7
o16 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (--x + -x x
7 1 2 2 4 1 5 1 7 2 3 2 7 1 2 1 2
-----------------------------------------------------------------------
32 3 1452 2 2 3 4 2 7 2 8 2
+ x x + 1, --x x + ----x x + 2x x + -x x x + -x x x + -x x x +
1 4 35 1 2 245 1 2 1 2 7 1 2 3 2 1 2 3 5 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
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 + 2x + 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 + 5x x - 2x x + 2x x x - x x x - x x x + 2x 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.