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
1 3 1 1 6 2 3
o3 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (-x + -x x +
5 1 7 2 4 1 5 1 2 2 3 2 5 1 7 1 2
------------------------------------------------------------------------
1 3 13 2 2 3 3 1 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 25 1 2 70 1 2 14 1 2 5 1 2 3 7 1 2 3 5 1 2 4
------------------------------------------------------------------------
1 2
-x x x + x x x x + 1), {x , x })
2 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 1
o6 = (map(R,R,{5x + -x + x , x , -x + -x + x , -x + 2x + x , x }),
1 9 2 5 1 9 1 2 2 4 5 1 2 3 2
------------------------------------------------------------------------
2 1 3 3 25 2 2 2 5 3
ideal (5x + -x x + x x - x , 125x x + --x x + 75x x x + --x x +
1 9 1 2 1 5 2 1 2 3 1 2 1 2 5 27 1 2
------------------------------------------------------------------------
10 2 2 1 4 1 3 1 2 2 3
--x x x + 15x x x + ---x + --x x + -x x + x x ), {x , x , x })
3 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} | 295245x_1x_2x_5^6-109350x_2^9x_5-5x_2^9+492075x
{-9} | 45x_1x_2^2x_5^3-4428675x_1x_2x_5^5+405x_1x_2x_5
{-9} | 45x_1x_2^3+4428675x_1x_2^2x_5^2+810x_1x_2^2x_5+
{-3} | 45x_1^2+x_1x_2+9x_1x_5-9x_2^3
------------------------------------------------------------------------
_2^8x_5^2+45x_2^8x_5-1476225x_2^7x_5^3-405x_2^7x_5^2+3645x_2^6x_5^3-
^4+1640250x_2^9-7381125x_2^8x_5-225x_2^8+22143375x_2^7x_5^2+4050x_2^
3177332285411250x_1x_2x_5^5-145282683375x_1x_2x_5^4+26572050x_1x_2x_
------------------------------------------------------------------------
32805x_2^5x_5^4+295245x_2^4x_5^5+6561x_2^2x_5^6+59049x_2x_5^7
7x_5-54675x_2^6x_5^2+492075x_2^5x_5^3-4428675x_2^4x_5^4+405x_2^4x_5
5^3+3645x_1x_2x_5^2-1176789735337500x_2^9+5295553809018750x_2^8x_5+
------------------------------------------------------------------------
^3+x_2^3x_5^3-98415x_2^2x_5^5+18x_2^2x_5^4-885735x_2x_5^6+81x_2x_5^5
242137805625x_2^8-15886661427056250x_2^7x_5^2-3632067084375x_2^7x_5+
------------------------------------------------------------------------
66430125x_2^7+39226324511250x_2^6x_5^2-1793613375x_2^6x_5-164025x_2^6-
------------------------------------------------------------------------
353036920601250x_2^5x_5^3+16142520375x_2^5x_5^2+1476225x_2^5x_5+405x_2^5
------------------------------------------------------------------------
+3177332285411250x_2^4x_5^4-145282683375x_2^4x_5^3+26572050x_2^4x_5^2+
------------------------------------------------------------------------
3645x_2^4x_5+x_2^4+98415x_2^3x_5^2+27x_2^3x_5+70607384120250x_2^2x_5^5-
------------------------------------------------------------------------
3228504075x_2^2x_5^4+1476225x_2^2x_5^3+243x_2^2x_5^2+635466457082250x_2x
------------------------------------------------------------------------
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_5^6-29056536675x_2x_5^5+5314410x_2x_5^4+729x_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
1 7 1 3 2
o13 = (map(R,R,{-x + 9x + x , x , --x + -x + x , x }), ideal (-x + 9x x
2 1 2 4 1 10 1 4 2 3 2 2 1 1 2
-----------------------------------------------------------------------
7 3 257 2 2 9 3 1 2 2 7 2
+ x x + 1, --x x + ---x x + -x x + -x x x + 9x x x + --x x x +
1 4 20 1 2 40 1 2 4 1 2 2 1 2 3 1 2 3 10 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
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
3 5 3 11 2
o16 = (map(R,R,{-x + 4x + x , x , -x + -x + x , x }), ideal (--x + 4x x
8 1 2 4 1 9 1 2 2 3 2 8 1 1 2
-----------------------------------------------------------------------
5 3 401 2 2 3 3 2 2 5 2
+ x x + 1, --x x + ---x x + 6x x + -x x x + 4x x x + -x x x +
1 4 24 1 2 144 1 2 1 2 8 1 2 3 1 2 3 9 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
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
2
o19 = (map(R,R,{x + x , x , 4x + 2x + x , x }), ideal (x + x x + x x +
2 4 1 1 2 3 2 1 1 2 1 4
-----------------------------------------------------------------------
2 2 3 2 2 2
1, 4x x + 2x x + x x x + 4x x x + 2x x x + x x x x + 1), {x ,
1 2 1 2 1 2 3 1 2 4 1 2 4 1 2 3 4 4
-----------------------------------------------------------------------
x })
3
o19 : Sequence
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This symbol is provided by the package NoetherNormalization.