Description
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
3 9 7 9 7 2
o3 = (map (R, R, {-x + -x + x , x , -x + -x + x , x }), ideal (-x +
4 1 2 2 4 1 4 1 4 2 3 2 4 1
------------------------------------------------------------------------
9 21 3 153 2 2 81 3 3 2 9 2
-x x + x x + 1, --x x + ---x x + --x x + -x x x + -x x x +
2 1 2 1 4 16 1 2 16 1 2 8 1 2 4 1 2 3 2 1 2 3
------------------------------------------------------------------------
7 2 9 2
-x x x + -x x x + x x x x + 1), {x , x })
4 1 2 4 4 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
7 7 7 3 7
o6 = (map (R, R, {-x + -x + x , x , 7x + --x + x , -x + --x + x , x }),
3 1 9 2 5 1 1 10 2 4 7 1 10 2 3 2
------------------------------------------------------------------------
7 2 7 3 343 3 343 2 2 49 2 343 3
ideal (-x + -x x + x x - x , ---x x + ---x x + --x x x + ---x x +
3 1 9 1 2 1 5 2 27 1 2 27 1 2 3 1 2 5 81 1 2
------------------------------------------------------------------------
98 2 2 343 4 49 3 7 2 2 3
--x x x + 7x x x + ---x + --x x + -x x + x x ), {x , x , x })
9 1 2 5 1 2 5 729 2 27 2 5 3 2 5 2 5 5 4 3
o6 : Sequence
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i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 413343x_1x_2x_5^6-3500658x_2^9x_5-117649x_2^9+2250423x_2^8x_5^2
{-9} | 7203x_1x_2^2x_5^3-137781x_1x_2x_5^5+9261x_1x_2x_5^4+1166886x_2^
{-9} | 121060821x_1x_2^3+2315685267x_1x_2^2x_5^2+311299254x_1x_2^2x_5+
{-3} | 21x_1^2+7x_1x_2+9x_1x_5-9x_2^3
------------------------------------------------------------------------
+151263x_2^8x_5-964467x_2^7x_5^3-194481x_2^7x_5^2+250047x_2^6x_5^3-
9-750141x_2^8x_5-16807x_2^8+321489x_2^7x_5^2+43218x_2^7x_5-83349x_2
439334834526x_1x_2x_5^5-14765025303x_1x_2x_5^4+1984873086x_1x_2x_5^
------------------------------------------------------------------------
321489x_2^5x_5^4+413343x_2^4x_5^5+137781x_2^2x_5^6+177147x_2x_5^
^6x_5^2+107163x_2^5x_5^3-137781x_2^4x_5^4+9261x_2^4x_5^3+2401x_2
3+200120949x_1x_2x_5^2-3720786376356x_2^9+2391934099086x_2^8x_5+
------------------------------------------------------------------------
7
^3x_5^3-45927x_2^2x_5^5+6174x_2^2x_5^4-59049x_2x_5^6+3969x_2x_
80387359983x_2^8-1025114613894x_2^7x_5^2-172258628535x_2^7x_5+
------------------------------------------------------------------------
5^5
2315685267x_2^7+265770455454x_2^6x_5^2-8931928887x_2^6x_5-600362847x_2^6
------------------------------------------------------------------------
-341704871298x_2^5x_5^3+11483908569x_2^5x_5^2+771895089x_2^5x_5+
------------------------------------------------------------------------
155649627x_2^5+439334834526x_2^4x_5^4-14765025303x_2^4x_5^3+1984873086x_
------------------------------------------------------------------------
2^4x_5^2+200120949x_2^4x_5+40353607x_2^4+771895089x_2^3x_5^2+155649627x_
------------------------------------------------------------------------
2^3x_5+146444944842x_2^2x_5^5-4921675101x_2^2x_5^4+1654060905x_2^2x_5^3+
------------------------------------------------------------------------
200120949x_2^2x_5^2+188286357654x_2x_5^6-6327867987x_2x_5^5+850659894x_
------------------------------------------------------------------------
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2x_5^4+85766121x_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
|
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 7 9 5 6 2
o13 = (map (R, R, {-x + -x + x , x , --x + -x + x , x }), ideal (-x +
5 1 4 2 4 1 10 1 6 2 3 2 5 1
-----------------------------------------------------------------------
7 9 3 209 2 2 35 3 1 2 7 2
-x x + x x + 1, --x x + ---x x + --x x + -x x x + -x x x +
4 1 2 1 4 50 1 2 120 1 2 24 1 2 5 1 2 3 4 1 2 3
-----------------------------------------------------------------------
9 2 5 2
--x x x + -x x x + x x x x + 1), {x , x })
10 1 2 4 6 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
7 2 7 10 10 2
o16 = (map (R, R, {-x + -x + x , x , -x + --x + x , x }), ideal (--x +
3 1 3 2 4 1 5 1 7 2 3 2 3 1
-----------------------------------------------------------------------
2 49 3 64 2 2 20 3 7 2 2 2
-x x + x x + 1, --x x + --x x + --x x + -x x x + -x x x +
3 1 2 1 4 15 1 2 15 1 2 21 1 2 3 1 2 3 3 1 2 3
-----------------------------------------------------------------------
7 2 10 2
-x x x + --x x x + x x x x + 1), {x , x })
5 1 2 4 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 - 2x + x , x , x - x + x , x }), ideal (- x -
1 2 4 1 1 2 3 2 1
-----------------------------------------------------------------------
3 3 2 2 2 2
2x x + x x + 1, - 2x x + 2x x - 2x x x - 2x x x + x x x - x x x
1 2 1 4 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.