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 6 9 8 11 2 6
o3 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (--x + -x x
5 1 7 2 4 1 7 1 7 2 3 2 5 1 7 1 2
------------------------------------------------------------------------
54 3 606 2 2 48 3 6 2 6 2 9 2
+ x x + 1, --x x + ---x x + --x x + -x x x + -x x x + -x x x +
1 4 35 1 2 245 1 2 49 1 2 5 1 2 3 7 1 2 3 7 1 2 4
------------------------------------------------------------------------
8 2
-x x x + x x x x + 1), {x , x })
7 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 1 1 7 7
o6 = (map(R,R,{-x + -x + x , x , x + -x + x , -x + -x + x , x }), ideal
7 1 2 2 5 1 1 2 2 4 9 1 9 2 3 2
------------------------------------------------------------------------
9 2 1 3 729 3 243 2 2 243 2 27 3
(-x + -x x + x x - x , ---x x + ---x x + ---x x x + --x x +
7 1 2 1 2 1 5 2 343 1 2 98 1 2 49 1 2 5 28 1 2
------------------------------------------------------------------------
27 2 27 2 1 4 3 3 3 2 2 3
--x x x + --x x x + -x + -x x + -x x + x x ), {x , x , x })
7 1 2 5 7 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} | 2016x_1x_2x_5^6-3888x_2^9x_5-63x_2^9+3888x_2^8x_5^
{-9} | 882x_1x_2^2x_5^3-54432x_1x_2x_5^5+1764x_1x_2x_5^4+
{-9} | 43218x_1x_2^3+2667168x_1x_2^2x_5^2+172872x_1x_2^2x
{-3} | 18x_1^2+7x_1x_2+14x_1x_5-14x_2^3
------------------------------------------------------------------------
2+126x_2^8x_5-2592x_2^7x_5^3-252x_2^7x_5^2+504x_2^6x_5^3-1008x_2^5x_5^4+
104976x_2^9-104976x_2^8x_5-1134x_2^8+69984x_2^7x_5^2+4536x_2^7x_5-13608x
_5+3386105856x_1x_2x_5^5-54867456x_1x_2x_5^4+3556224x_1x_2x_5^3+172872x_
------------------------------------------------------------------------
2016x_2^4x_5^5+784x_2^2x_5^6+1568x_2x_5^7
_2^6x_5^2+27216x_2^5x_5^3-54432x_2^4x_5^4+1764x_2^4x_5^3+343x_2^3x_5^3-
1x_2x_5^2-6530347008x_2^9+6530347008x_2^8x_5+105815808x_2^8-4353564672x
------------------------------------------------------------------------
21168x_2^2x_5^5+1372x_2^2x_5^4-42336x_2x_5^6+1372x_2x_5^5
_2^7x_5^2-352719360x_2^7x_5+2286144x_2^7+846526464x_2^6x_5^2-13716864x_2
------------------------------------------------------------------------
^6x_5-444528x_2^6-1693052928x_2^5x_5^3+27433728x_2^5x_5^2+889056x_2^5x_5
------------------------------------------------------------------------
+86436x_2^5+3386105856x_2^4x_5^4-54867456x_2^4x_5^3+3556224x_2^4x_5^2+
------------------------------------------------------------------------
172872x_2^4x_5+16807x_2^4+1037232x_2^3x_5^2+100842x_2^3x_5+1316818944x_2
------------------------------------------------------------------------
^2x_5^5-21337344x_2^2x_5^4+3457440x_2^2x_5^3+201684x_2^2x_5^2+
------------------------------------------------------------------------
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2633637888x_2x_5^6-42674688x_2x_5^5+2765952x_2x_5^4+134456x_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
4 2 11 2
o13 = (map(R,R,{-x + 5x + x , x , -x + 4x + x , x }), ideal (--x + 5x x
7 1 2 4 1 5 1 2 3 2 7 1 1 2
-----------------------------------------------------------------------
8 3 30 2 2 3 4 2 2 2 2
+ x x + 1, --x x + --x x + 20x x + -x x x + 5x x x + -x x x +
1 4 35 1 2 7 1 2 1 2 7 1 2 3 1 2 3 5 1 2 4
-----------------------------------------------------------------------
2
4x x x + x x x x + 1), {x , x })
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
10 7 7 2 13 2
o16 = (map(R,R,{--x + -x + x , x , -x + -x + x , x }), ideal (--x +
3 1 2 2 4 1 4 1 5 2 3 2 3 1
-----------------------------------------------------------------------
7 35 3 179 2 2 7 3 10 2 7 2
-x x + x x + 1, --x x + ---x x + -x x + --x x x + -x x x +
2 1 2 1 4 6 1 2 24 1 2 5 1 2 3 1 2 3 2 1 2 3
-----------------------------------------------------------------------
7 2 2 2
-x x x + -x x x + x x x x + 1), {x , x })
4 1 2 4 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
--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,{2x + 6x + x , x , - 4x - 2x + x , x }), ideal (3x +
1 2 4 1 1 2 3 2 1
-----------------------------------------------------------------------
3 2 2 3 2 2
6x x + x x + 1, - 8x x - 28x x - 12x x + 2x x x + 6x x x -
1 2 1 4 1 2 1 2 1 2 1 2 3 1 2 3
-----------------------------------------------------------------------
2 2
4x 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.