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
5 1 8 2 1
o3 = (map(R,R,{-x + -x + x , x , 2x + 2x + x , x }), ideal (-x + -x x +
3 1 3 2 4 1 1 2 3 2 3 1 3 1 2
------------------------------------------------------------------------
10 3 2 2 2 3 5 2 1 2 2 2
x x + 1, --x x + 4x x + -x x + -x x x + -x x x + 2x x x + 2x x x
1 4 3 1 2 1 2 3 1 2 3 1 2 3 3 1 2 3 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
1 9 7 1 2
o6 = (map(R,R,{-x + -x + x , x , x + -x + x , -x + -x + x , x }), ideal
2 1 4 2 5 1 1 3 2 4 2 1 9 2 3 2
------------------------------------------------------------------------
1 2 9 3 1 3 27 2 2 3 2 243 3 27 2
(-x + -x x + x x - x , -x x + --x x + -x x x + ---x x + --x x x
2 1 4 1 2 1 5 2 8 1 2 16 1 2 4 1 2 5 32 1 2 4 1 2 5
------------------------------------------------------------------------
3 2 729 4 243 3 27 2 2 3
+ -x x x + ---x + ---x x + --x x + x x ), {x , x , x })
2 1 2 5 64 2 16 2 5 4 2 5 2 5 5 4 3
o6 : Sequence
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i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 1024x_1x_2x_5^6-15552x_2^9x_5-59049x_2^9+3456x_2^8x_5^2+26244x_2
{-9} | 4374x_1x_2^2x_5^3-256x_1x_2x_5^5+1944x_1x_2x_5^4+3888x_2^9-864x_
{-9} | 6973568802x_1x_2^3+408146688x_1x_2^2x_5^2+6198727824x_1x_2^2x_5+
{-3} | 2x_1^2+9x_1x_2+4x_1x_5-4x_2^3
------------------------------------------------------------------------
^8x_5-512x_2^7x_5^3-11664x_2^7x_5^2+5184x_2^6x_5^3-2304x_2^5x_5^4+1024x
2^8x_5-2187x_2^8+128x_2^7x_5^2+1944x_2^7x_5-1296x_2^6x_5^2+576x_2^5x_5^
2097152x_1x_2x_5^5-7962624x_1x_2x_5^4+120932352x_1x_2x_5^3+1377495072x_
------------------------------------------------------------------------
_2^4x_5^5+4608x_2^2x_5^6+2048x_2x_5^7
3-256x_2^4x_5^4+1944x_2^4x_5^3+19683x_2^3x_5^3-1152x_2^2x_5^5+17496x_2^
1x_2x_5^2-31850496x_2^9+7077888x_2^8x_5+26873856x_2^8-1048576x_2^7x_5^2
------------------------------------------------------------------------
2x_5^4-512x_2x_5^6+3888x_2x_5^5
-19906560x_2^7x_5+30233088x_2^7+10616832x_2^6x_5^2-40310784x_2^6x_5-
------------------------------------------------------------------------
306110016x_2^6-4718592x_2^5x_5^3+17915904x_2^5x_5^2+136048896x_2^5x_5+
------------------------------------------------------------------------
3099363912x_2^5+2097152x_2^4x_5^4-7962624x_2^4x_5^3+120932352x_2^4x_5^2+
------------------------------------------------------------------------
1377495072x_2^4x_5+31381059609x_2^4+1836660096x_2^3x_5^2+41841412812x_2^
------------------------------------------------------------------------
3x_5+9437184x_2^2x_5^5-35831808x_2^2x_5^4+1360488960x_2^2x_5^3+
------------------------------------------------------------------------
18596183472x_2^2x_5^2+4194304x_2x_5^6-15925248x_2x_5^5+241864704x_2x_5^4
------------------------------------------------------------------------
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+2754990144x_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 1 4 4 7 2 1
o13 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (-x + -x x
4 1 5 2 4 1 3 1 3 2 3 2 4 1 5 1 2
-----------------------------------------------------------------------
3 19 2 2 4 3 3 2 1 2 4 2
+ x x + 1, x x + --x x + --x x + -x x x + -x x x + -x x x +
1 4 1 2 15 1 2 15 1 2 4 1 2 3 5 1 2 3 3 1 2 4
-----------------------------------------------------------------------
4 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
1 5 4 5 2
o16 = (map(R,R,{-x + 5x + x , x , -x + -x + x , x }), ideal (-x + 5x x
4 1 2 4 1 7 1 5 2 3 2 4 1 1 2
-----------------------------------------------------------------------
5 3 132 2 2 3 1 2 2 5 2
+ x x + 1, --x x + ---x x + 4x x + -x x x + 5x x x + -x x x +
1 4 28 1 2 35 1 2 1 2 4 1 2 3 1 2 3 7 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
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 - 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 2
x x + 1, 4x x - 6x x + 2x x + 2x x x - 2x x x + 2x 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.