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
7 3 12 2 3
o3 = (map(R,R,{-x + -x + x , x , 5x + x + x , x }), ideal (--x + -x x +
5 1 2 2 4 1 1 2 3 2 5 1 2 1 2
------------------------------------------------------------------------
3 89 2 2 3 3 7 2 3 2 2 2
x x + 1, 7x x + --x x + -x x + -x x x + -x x x + 5x x x + x x x
1 4 1 2 10 1 2 2 1 2 5 1 2 3 2 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 2 7 1 4
o6 = (map(R,R,{x + -x + x , x , -x + -x + x , -x + -x + x , x }), ideal
1 7 2 5 1 3 1 6 2 4 2 1 3 2 3 2
------------------------------------------------------------------------
2 1 3 3 3 2 2 2 3 3 6 2
(x + -x x + x x - x , x x + -x x + 3x x x + --x x + -x x x +
1 7 1 2 1 5 2 1 2 7 1 2 1 2 5 49 1 2 7 1 2 5
------------------------------------------------------------------------
2 1 4 3 3 3 2 2 3
3x x x + ---x + --x x + -x x + x x ), {x , x , x })
1 2 5 343 2 49 2 5 7 2 5 2 5 5 4 3
o6 : Sequence
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i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 16807x_1x_2x_5^6-2058x_2^9x_5-x_2^9+7203x_2^8x_5^2+7x_2^
{-9} | 7x_1x_2^2x_5^3-50421x_1x_2x_5^5+49x_1x_2x_5^4+6174x_2^9-
{-9} | 7x_1x_2^3+50421x_1x_2^2x_5^2+98x_1x_2^2x_5+249143169618x
{-3} | 7x_1^2+x_1x_2+7x_1x_5-7x_2^3
------------------------------------------------------------------------
8x_5-16807x_2^7x_5^3-49x_2^7x_5^2+343x_2^6x_5^3-2401x_2^5x_5^4+
21609x_2^8x_5-7x_2^8+50421x_2^7x_5^2+98x_2^7x_5-1029x_2^6x_5^2+
_1x_2x_5^5-121060821x_1x_2x_5^4+235298x_1x_2x_5^3+343x_1x_2x_5^
------------------------------------------------------------------------
16807x_2^4x_5^5+2401x_2^2x_5^6+16807x_2x_5^7
7203x_2^5x_5^3-50421x_2^4x_5^4+49x_2^4x_5^3+x_2^3x_5^3-7203x_2^2x_5^5+
2-30507326892x_2^9+106775644122x_2^8x_5+51883209x_2^8-249143169618x_2^
------------------------------------------------------------------------
14x_2^2x_5^4-50421x_2x_5^6+49x_2x_5^5
7x_5^2-605304105x_2^7x_5+117649x_2^7+5084554482x_2^6x_5^2-2470629x_2^6x_
------------------------------------------------------------------------
5-2401x_2^6-35591881374x_2^5x_5^3+17294403x_2^5x_5^2+16807x_2^5x_5+49x_2
------------------------------------------------------------------------
^5+249143169618x_2^4x_5^4-121060821x_2^4x_5^3+235298x_2^4x_5^2+343x_2^4x
------------------------------------------------------------------------
_5+x_2^4+7203x_2^3x_5^2+21x_2^3x_5+35591881374x_2^2x_5^5-17294403x_2^2x_
------------------------------------------------------------------------
5^4+84035x_2^2x_5^3+147x_2^2x_5^2+249143169618x_2x_5^6-121060821x_2x_5^5
------------------------------------------------------------------------
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+235298x_2x_5^4+343x_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
5 9 8 10 7 2 9
o13 = (map(R,R,{-x + -x + x , x , -x + --x + x , x }), ideal (-x + -x x
2 1 5 2 4 1 3 1 9 2 3 2 2 1 5 1 2
-----------------------------------------------------------------------
20 3 341 2 2 3 5 2 9 2 8 2
+ x x + 1, --x x + ---x x + 2x x + -x x x + -x x x + -x x x +
1 4 3 1 2 45 1 2 1 2 2 1 2 3 5 1 2 3 3 1 2 4
-----------------------------------------------------------------------
10 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
7 3 16 2 3
o16 = (map(R,R,{-x + -x + x , x , 3x + x + x , x }), ideal (--x + -x x
9 1 4 2 4 1 1 2 3 2 9 1 4 1 2
-----------------------------------------------------------------------
7 3 109 2 2 3 3 7 2 3 2 2
+ x x + 1, -x x + ---x x + -x x + -x x x + -x x x + 3x x x +
1 4 3 1 2 36 1 2 4 1 2 9 1 2 3 4 1 2 3 1 2 4
-----------------------------------------------------------------------
2
x x x + x x x x + 1), {x , x })
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,{- 4x + 3x + x , x , 2x - 3x + x , x }), ideal (- 3x +
1 2 4 1 1 2 3 2 1
-----------------------------------------------------------------------
3 2 2 3 2 2
3x x + x x + 1, - 8x x + 18x x - 9x x - 4x x x + 3x x x +
1 2 1 4 1 2 1 2 1 2 1 2 3 1 2 3
-----------------------------------------------------------------------
2 2
2x x x - 3x 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.