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
6 2
o3 = (map(R,R,{x + 4x + x , x , 2x + -x + x , x }), ideal (2x + 4x x +
1 2 4 1 1 7 2 3 2 1 1 2
------------------------------------------------------------------------
3 62 2 2 24 3 2 2 2 6 2
x x + 1, 2x x + --x x + --x x + x x x + 4x x x + 2x x x + -x x x
1 4 1 2 7 1 2 7 1 2 1 2 3 1 2 3 1 2 4 7 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
4 9 5 1 1
o6 = (map(R,R,{-x + -x + x , x , 4x + -x + x , -x + -x + x , x }),
9 1 2 2 5 1 1 9 2 4 9 1 9 2 3 2
------------------------------------------------------------------------
4 2 9 3 64 3 8 2 2 16 2 3
ideal (-x + -x x + x x - x , ---x x + -x x + --x x x + 27x x +
9 1 2 1 2 1 5 2 729 1 2 3 1 2 27 1 2 5 1 2
------------------------------------------------------------------------
2 4 2 729 4 243 3 27 2 2 3
12x x x + -x x x + ---x + ---x x + --x x + x x ), {x , x , x })
1 2 5 3 1 2 5 8 2 4 2 5 2 2 5 2 5 5 4 3
o6 : Sequence
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i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 288x_1x_2x_5^6-15552x_2^9x_5-531441x_2^9+1728x_2
{-9} | 1417176x_1x_2^2x_5^3-4608x_1x_2x_5^5+314928x_1x_
{-9} | 183014339639688x_1x_2^3+595077871104x_1x_2^2x_5^
{-3} | 8x_1^2+81x_1x_2+18x_1x_5-18x_2^3
------------------------------------------------------------------------
^8x_5^2+118098x_2^8x_5-128x_2^7x_5^3-26244x_2^7x_5^2+5832x_2^6x_5^3-
2x_5^4+248832x_2^9-27648x_2^8x_5-629856x_2^8+2048x_2^7x_5^2+279936x_
2+81339706506528x_1x_2^2x_5+18874368x_1x_2x_5^5-644972544x_1x_2x_5^4
------------------------------------------------------------------------
1296x_2^5x_5^4+288x_2^4x_5^5+2916x_2^2x_5^6+648x_2x_5^7
2^7x_5-93312x_2^6x_5^2+20736x_2^5x_5^3-4608x_2^4x_5^4+314928x_2^
+88159684608x_1x_2x_5^3+9037745167392x_1x_2x_5^2-1019215872x_2^9
------------------------------------------------------------------------
4x_5^3+14348907x_2^3x_5^3-46656x_2^2x_5^5+6377292x_2^2x_5^4-10368x_2x_5^
+113246208x_2^8x_5+3869835264x_2^8-8388608x_2^7x_5^2-1433272320x_2^7x_5+
------------------------------------------------------------------------
6+708588x_2x_5^5
19591041024x_2^7+382205952x_2^6x_5^2-13060694016x_2^6x_5-892616806656x_2
------------------------------------------------------------------------
^6-84934656x_2^5x_5^3+2902376448x_2^5x_5^2+198359290368x_2^5x_5+
------------------------------------------------------------------------
40669853253264x_2^5+18874368x_2^4x_5^4-644972544x_2^4x_5^3+88159684608x_
------------------------------------------------------------------------
2^4x_5^2+9037745167392x_2^4x_5+1853020188851841x_2^4+6025163444928x_2^3x
------------------------------------------------------------------------
_5^2+1235346792567894x_2^3x_5+191102976x_2^2x_5^5-6530347008x_2^2x_5^4+
------------------------------------------------------------------------
2231542016640x_2^2x_5^3+274521509459532x_2^2x_5^2+42467328x_2x_5^6-
------------------------------------------------------------------------
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1451188224x_2x_5^5+198359290368x_2x_5^4+20334926626632x_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
2 3 7 2
o13 = (map(R,R,{-x + 10x + x , x , -x + 4x + x , x }), ideal (-x +
5 1 2 4 1 5 1 2 3 2 5 1
-----------------------------------------------------------------------
6 3 38 2 2 3 2 2 2
10x x + x x + 1, --x x + --x x + 40x x + -x x x + 10x x x +
1 2 1 4 25 1 2 5 1 2 1 2 5 1 2 3 1 2 3
-----------------------------------------------------------------------
3 2 2
-x x x + 4x x x + x x x x + 1), {x , x })
5 1 2 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
7 2
o16 = (map(R,R,{x + x + x , x , x + -x + x , x }), ideal (2x + x x +
1 2 4 1 1 2 2 3 2 1 1 2
-----------------------------------------------------------------------
3 9 2 2 7 3 2 2 2 7 2
x x + 1, x x + -x x + -x x + x x x + x x x + x x x + -x x x +
1 4 1 2 2 1 2 2 1 2 1 2 3 1 2 3 1 2 4 2 1 2 4
-----------------------------------------------------------------------
x x x x + 1), {x , x })
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 + x , x , 2x + 2x + x , x }), ideal (x + 2x x + x x
2 4 1 1 2 3 2 1 1 2 1 4
-----------------------------------------------------------------------
2 2 3 2 2 2
+ 1, 4x x + 4x x + 2x x x + 2x 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.