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 2 7 11 2 2
o3 = (map(R,R,{-x + -x + x , x , -x + 2x + x , x }), ideal (--x + -x x
6 1 5 2 4 1 4 1 2 3 2 6 1 5 1 2
------------------------------------------------------------------------
35 3 71 2 2 4 3 5 2 2 2 7 2
+ x x + 1, --x x + --x x + -x x + -x x x + -x x x + -x x x +
1 4 24 1 2 30 1 2 5 1 2 6 1 2 3 5 1 2 3 4 1 2 4
------------------------------------------------------------------------
2
2x x x + x x x x + 1), {x , x })
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
9 5 1 5 1
o6 = (map(R,R,{-x + -x + x , x , -x + 3x + x , -x + -x + x , x }),
5 1 4 2 5 1 4 1 2 4 4 1 9 2 3 2
------------------------------------------------------------------------
9 2 5 3 729 3 243 2 2 243 2 135 3
ideal (-x + -x x + x x - x , ---x x + ---x x + ---x x x + ---x x
5 1 4 1 2 1 5 2 125 1 2 20 1 2 25 1 2 5 16 1 2
------------------------------------------------------------------------
27 2 27 2 125 4 75 3 15 2 2 3
+ --x x x + --x x x + ---x + --x x + --x x + x x ), {x , x , x })
2 1 2 5 5 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} | 46080x_1x_2x_5^6-777600x_2^9x_5-140625x
{-9} | 562500x_1x_2^2x_5^3-1244160x_1x_2x_5^5+
{-9} | 1098632812500x_1x_2^3+2430000000000x_1x
{-3} | 36x_1^2+25x_1x_2+20x_1x_5-20x_2^3
------------------------------------------------------------------------
_2^9+311040x_2^8x_5^2+112500x_2^8x_5-82944x_2^7x_5^3-90000x_
450000x_1x_2x_5^4+20995200x_2^9-8398080x_2^8x_5-1012500x_2^8
_2^2x_5^2+1757812500000x_1x_2^2x_5+9906778275840x_1x_2x_5^5-
------------------------------------------------------------------------
2^7x_5^2+72000x_2^6x_5^3-57600x_2^5x_5^4+46080x_2^4x_5^5+32000x_2^2x_5^
+2239488x_2^7x_5^2+1620000x_2^7x_5-1944000x_2^6x_5^2+1555200x_2^5x_5^3-
1791590400000x_1x_2x_5^4+1296000000000x_1x_2x_5^3+703125000000x_1x_2x_5
------------------------------------------------------------------------
6+25600x_2x_5^7
1244160x_2^4x_5^4+450000x_2^4x_5^3+390625x_2^3x_5^3-864000x_2^2x_5^
^2-167176883404800x_2^9+66870753361920x_2^8x_5+12093235200000x_2^8-
------------------------------------------------------------------------
5+625000x_2^2x_5^4-691200x_2x_5^6+250000x_2x_5^5
17832200896512x_2^7x_5^2-16124313600000x_2^7x_5+1166400000000x_2^7+
------------------------------------------------------------------------
15479341056000x_2^6x_5^2-2799360000000x_2^6x_5-1012500000000x_2^6-
------------------------------------------------------------------------
12383472844800x_2^5x_5^3+2239488000000x_2^5x_5^2+810000000000x_2^5x_5+
------------------------------------------------------------------------
878906250000x_2^5+9906778275840x_2^4x_5^4-1791590400000x_2^4x_5^3+
------------------------------------------------------------------------
1296000000000x_2^4x_5^2+703125000000x_2^4x_5+762939453125x_2^4+
------------------------------------------------------------------------
1687500000000x_2^3x_5^2+1831054687500x_2^3x_5+6879707136000x_2^2x_5^5-
------------------------------------------------------------------------
1244160000000x_2^2x_5^4+2250000000000x_2^2x_5^3+1464843750000x_2^2x_5^2+
------------------------------------------------------------------------
5503765708800x_2x_5^6-995328000000x_2x_5^5+720000000000x_2x_5^4+
------------------------------------------------------------------------
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390625000000x_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
|
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
2 6 5 2 2
o13 = (map(R,R,{x + -x + x , x , -x + -x + x , x }), ideal (2x + -x x +
1 9 2 4 1 7 1 6 2 3 2 1 9 1 2
-----------------------------------------------------------------------
6 3 43 2 2 5 3 2 2 2 6 2
x x + 1, -x x + --x x + --x x + x x x + -x x x + -x x x +
1 4 7 1 2 42 1 2 27 1 2 1 2 3 9 1 2 3 7 1 2 4
-----------------------------------------------------------------------
5 2
-x x x + x x x x + 1), {x , x })
6 1 2 4 1 2 3 4 4 3
o13 : Sequence
|
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 7 3 2
o16 = (map(R,R,{-x + 2x + x , x , -x + x + x , x }), ideal (-x + 2x x +
2 1 2 4 1 3 1 2 3 2 2 1 1 2
-----------------------------------------------------------------------
7 3 31 2 2 3 1 2 2 7 2 2
x x + 1, -x x + --x x + 2x x + -x x x + 2x x x + -x x x + x x x
1 4 6 1 2 6 1 2 1 2 2 1 2 3 1 2 3 3 1 2 4 1 2 4
-----------------------------------------------------------------------
+ x x x x + 1), {x , x })
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
--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: 3
--trying with basis element limit: 5
--trying with basis element limit: 20
2
o19 = (map(R,R,{- 5x + 2x + x , x , 2x - 2x + x , x }), ideal (- 4x +
1 2 4 1 1 2 3 2 1
-----------------------------------------------------------------------
3 2 2 3 2 2
2x x + x x + 1, - 10x x + 14x x - 4x x - 5x x x + 2x x x +
1 2 1 4 1 2 1 2 1 2 1 2 3 1 2 3
-----------------------------------------------------------------------
2 2
2x x x - 2x 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.