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 7 7 9 14 2 7
o3 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (--x + -x x
9 1 9 2 4 1 9 1 4 2 3 2 9 1 9 1 2
------------------------------------------------------------------------
35 3 601 2 2 7 3 5 2 7 2 7 2
+ x x + 1, --x x + ---x x + -x x + -x x x + -x x x + -x x x +
1 4 81 1 2 324 1 2 4 1 2 9 1 2 3 9 1 2 3 9 1 2 4
------------------------------------------------------------------------
9 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
9 3 1
o6 = (map(R,R,{-x + -x + x , x , x + 2x + x , -x + 7x + x , x }), ideal
8 1 4 2 5 1 1 2 4 4 1 2 3 2
------------------------------------------------------------------------
9 2 3 3 729 3 729 2 2 243 2 243 3
(-x + -x x + x x - x , ---x x + ---x x + ---x x x + ---x x +
8 1 4 1 2 1 5 2 512 1 2 256 1 2 64 1 2 5 128 1 2
------------------------------------------------------------------------
81 2 27 2 27 4 27 3 9 2 2 3
--x x x + --x x x + --x + --x x + -x x + x x ), {x , x , x })
16 1 2 5 8 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} | 9216x_1x_2x_5^6-34992x_2^9x_5-2187x_2^9+23328x_2^8x_5^2+2916x_2
{-9} | 216x_1x_2^2x_5^3-2304x_1x_2x_5^5+288x_1x_2x_5^4+8748x_2^9-5832x
{-9} | 243x_1x_2^3+2592x_1x_2^2x_5^2+648x_1x_2^2x_5+147456x_1x_2x_5^5-
{-3} | 9x_1^2+6x_1x_2+8x_1x_5-8x_2^3
------------------------------------------------------------------------
^8x_5-10368x_2^7x_5^3-3888x_2^7x_5^2+5184x_2^6x_5^3-6912x_2^5x_5^4+9216x
_2^8x_5-243x_2^8+2592x_2^7x_5^2+648x_2^7x_5-1296x_2^6x_5^2+1728x_2^5x_5^
9216x_1x_2x_5^4+2304x_1x_2x_5^3+432x_1x_2x_5^2-559872x_2^9+373248x_2^8x_
------------------------------------------------------------------------
_2^4x_5^5+6144x_2^2x_5^6+8192x_2x_5^7
3-2304x_2^4x_5^4+288x_2^4x_5^3+144x_2^3x_5^3-1536x_2^2x_5^5+384x_2^2x_
5+23328x_2^8-165888x_2^7x_5^2-51840x_2^7x_5+1296x_2^7+82944x_2^6x_5^2-
------------------------------------------------------------------------
5^4-2048x_2x_5^6+256x_2x_5^5
5184x_2^6x_5-648x_2^6-110592x_2^5x_5^3+6912x_2^5x_5^2+864x_2^5x_5+324x_2
------------------------------------------------------------------------
^5+147456x_2^4x_5^4-9216x_2^4x_5^3+2304x_2^4x_5^2+432x_2^4x_5+162x_2^4+
------------------------------------------------------------------------
1728x_2^3x_5^2+648x_2^3x_5+98304x_2^2x_5^5-6144x_2^2x_5^4+3840x_2^2x_5^3
------------------------------------------------------------------------
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+864x_2^2x_5^2+131072x_2x_5^6-8192x_2x_5^5+2048x_2x_5^4+384x_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
--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 37 31 7 2
o13 = (map(R,R,{-x + --x + x , x , 2x + --x + x , x }), ideal (-x +
5 1 20 2 4 1 1 12 2 3 2 5 1
-----------------------------------------------------------------------
37 4 3 71 2 2 1147 3 2 2 37 2
--x x + x x + 1, -x x + --x x + ----x x + -x x x + --x x x +
20 1 2 1 4 5 1 2 15 1 2 240 1 2 5 1 2 3 20 1 2 3
-----------------------------------------------------------------------
2 31 2
2x x x + --x x x + x x x x + 1), {x , x })
1 2 4 12 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
8 9 2 2 8
o16 = (map(R,R,{2x + -x + x , x , -x + -x + x , x }), ideal (3x + -x x
1 5 2 4 1 4 1 5 2 3 2 1 5 1 2
-----------------------------------------------------------------------
9 3 22 2 2 16 3 2 8 2 9 2
+ x x + 1, -x x + --x x + --x x + 2x x x + -x x x + -x x x +
1 4 2 1 2 5 1 2 25 1 2 1 2 3 5 1 2 3 4 1 2 4
-----------------------------------------------------------------------
2 2
-x x x + x x x x + 1), {x , x })
5 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
2
o19 = (map(R,R,{x + x , x , 2x - x + 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, 2x x - x x + x x x + 2x x x - x x x + x x x x + 1), {x , x })
1 2 1 2 1 2 3 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.