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 1 9 2
o3 = (map(R,R,{-x + 2x + x , x , 4x + -x + x , x }), ideal (-x + 2x x +
5 1 2 4 1 1 4 2 3 2 5 1 1 2
------------------------------------------------------------------------
16 3 41 2 2 1 3 4 2 2 2
x x + 1, --x x + --x x + -x x + -x x x + 2x x x + 4x x x +
1 4 5 1 2 5 1 2 2 1 2 5 1 2 3 1 2 3 1 2 4
------------------------------------------------------------------------
1 2
-x x x + x x x x + 1), {x , x })
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
1 7 9 7 9
o6 = (map(R,R,{-x + -x + x , x , --x + x + x , --x + -x + x , x }),
3 1 9 2 5 1 10 1 2 4 10 1 2 2 3 2
------------------------------------------------------------------------
1 2 7 3 1 3 7 2 2 1 2 49 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
------------------------------------------------------------------------
14 2 2 343 4 49 3 7 2 2 3
--x x x + x 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} | 59049x_1x_2x_5^6-71442x_2^9x_5-16807x_2^9+45927x_2^8x_5^2
{-9} | 7203x_1x_2^2x_5^3-19683x_1x_2x_5^5+9261x_1x_2x_5^4+23814x
{-9} | 5931980229x_1x_2^3+16209796869x_1x_2^2x_5^2+15253663446x_
{-3} | 3x_1^2+7x_1x_2+9x_1x_5-9x_2^3
------------------------------------------------------------------------
+21609x_2^8x_5-19683x_2^7x_5^3-27783x_2^7x_5^2+35721x_2^6x_5^3-45927x_2^
_2^9-15309x_2^8x_5-2401x_2^8+6561x_2^7x_5^2+6174x_2^7x_5-11907x_2^6x_5^2
1x_2^2x_5+62762119218x_1x_2x_5^5-14765025303x_1x_2x_5^4+13894111602x_1x_
------------------------------------------------------------------------
5x_5^4+59049x_2^4x_5^5+137781x_2^2x_5^6+177147x_2x_5^7
+15309x_2^5x_5^3-19683x_2^4x_5^4+9261x_2^4x_5^3+16807x_2^3x_5^3-
2x_5^3+9805926501x_1x_2x_5^2-75934415844x_2^9+48814981614x_2^8x_
------------------------------------------------------------------------
45927x_2^2x_5^5+43218x_2^2x_5^4-59049x_2x_5^6+27783x_2x_5^5
5+11483908569x_2^8-20920706406x_2^7x_5^2-24608375505x_2^7x_5+2315685267x
------------------------------------------------------------------------
_2^7+37967207922x_2^6x_5^2-8931928887x_2^6x_5-4202539929x_2^6-
------------------------------------------------------------------------
48814981614x_2^5x_5^3+11483908569x_2^5x_5^2+5403265623x_2^5x_5+
------------------------------------------------------------------------
7626831723x_2^5+62762119218x_2^4x_5^4-14765025303x_2^4x_5^3+13894111602x
------------------------------------------------------------------------
_2^4x_5^2+9805926501x_2^4x_5+13841287201x_2^4+37822859361x_2^3x_5^2+
------------------------------------------------------------------------
53387822061x_2^3x_5+146444944842x_2^2x_5^5-34451725707x_2^2x_5^4+
------------------------------------------------------------------------
81048984345x_2^2x_5^3+68641485507x_2^2x_5^2+188286357654x_2x_5^6-
------------------------------------------------------------------------
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44295075909x_2x_5^5+41682334806x_2x_5^4+29417779503x_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 3 3 2
o13 = (map(R,R,{-x + 9x + x , x , x + -x + x , x }), ideal (-x + 9x x +
2 1 2 4 1 1 2 2 3 2 2 1 1 2
-----------------------------------------------------------------------
1 3 39 2 2 27 3 1 2 2 2
x x + 1, -x x + --x x + --x x + -x x x + 9x x x + x x x +
1 4 2 1 2 4 1 2 2 1 2 2 1 2 3 1 2 3 1 2 4
-----------------------------------------------------------------------
3 2
-x x x + x x x x + 1), {x , x })
2 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
2 1 4 2 2
o16 = (map(R,R,{2x + -x + x , x , -x + -x + x , x }), ideal (3x + -x x
1 3 2 4 1 3 1 3 2 3 2 1 3 1 2
-----------------------------------------------------------------------
2 3 26 2 2 8 3 2 2 2 1 2
+ x x + 1, -x x + --x x + -x x + 2x x x + -x x x + -x x x +
1 4 3 1 2 9 1 2 9 1 2 1 2 3 3 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
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,{- 4x - 4x + x , x , 6x + 4x + x , x }), ideal (- 3x -
1 2 4 1 1 2 3 2 1
-----------------------------------------------------------------------
3 2 2 3 2 2
4x x + x x + 1, - 24x x - 40x x - 16x x - 4x x x - 4x x x +
1 2 1 4 1 2 1 2 1 2 1 2 3 1 2 3
-----------------------------------------------------------------------
2 2
6x x x + 4x x x + x x x x + 1), {x , x })
1 2 4 1 2 4 1 2 3 4 4 3
o19 : Sequence
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This symbol is provided by the package NoetherNormalization.