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 5 1 13 2 5
o3 = (map(R,R,{-x + -x + x , x , x + -x + x , x }), ideal (--x + -x x +
7 1 9 2 4 1 1 7 2 3 2 7 1 9 1 2
------------------------------------------------------------------------
6 3 299 2 2 5 3 6 2 5 2 2
x x + 1, -x x + ---x x + --x x + -x x x + -x x x + x x x +
1 4 7 1 2 441 1 2 63 1 2 7 1 2 3 9 1 2 3 1 2 4
------------------------------------------------------------------------
1 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
1 7 1 9 10
o6 = (map(R,R,{-x + -x + x , x , -x + 2x + x , -x + --x + x , x }),
3 1 4 2 5 1 5 1 2 4 5 1 7 2 3 2
------------------------------------------------------------------------
1 2 7 3 1 3 7 2 2 1 2 49 3
ideal (-x + -x x + x x - x , --x x + --x x + -x x x + --x x +
3 1 4 1 2 1 5 2 27 1 2 12 1 2 3 1 2 5 16 1 2
------------------------------------------------------------------------
7 2 2 343 4 147 3 21 2 2 3
-x x x + x x x + ---x + ---x x + --x x + x x ), {x , x , x })
2 1 2 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} | 3072x_1x_2x_5^6-18816x_2^9x_5-50421x_2^9+5376x_2^8x_5^2+28812x
{-9} | 28812x_1x_2^2x_5^3-3072x_1x_2x_5^5+16464x_1x_2x_5^4+18816x_2^9
{-9} | 71183762748x_1x_2^3+7589772288x_1x_2^2x_5^2+81352871712x_1x_2^
{-3} | 4x_1^2+21x_1x_2+12x_1x_5-12x_2^3
------------------------------------------------------------------------
_2^8x_5-1024x_2^7x_5^3-16464x_2^7x_5^2+9408x_2^6x_5^3-5376x_2^5x_5^4
-5376x_2^8x_5-9604x_2^8+1024x_2^7x_5^2+10976x_2^7x_5-9408x_2^6x_5^2+
2x_5+100663296x_1x_2x_5^5-269746176x_1x_2x_5^4+2891341824x_1x_2x_5^3
------------------------------------------------------------------------
+3072x_2^4x_5^5+16128x_2^2x_5^6+9216x_2x_5^7
5376x_2^5x_5^3-3072x_2^4x_5^4+16464x_2^4x_5^3+151263x_2^3x_5^3-16128x_2^
+23243677632x_1x_2x_5^2-616562688x_2^9+176160768x_2^8x_5+472055808x_2^8-
------------------------------------------------------------------------
2x_5^5+172872x_2^2x_5^4-9216x_2x_5^6+49392x_2x_5^5
33554432x_2^7x_5^2-449576960x_2^7x_5+481890304x_2^7+308281344x_2^6x_5^2-
------------------------------------------------------------------------
826097664x_2^6x_5-4427367168x_2^6-176160768x_2^5x_5^3+472055808x_2^5x_5^
------------------------------------------------------------------------
2+2529924096x_2^5x_5+40676435856x_2^5+100663296x_2^4x_5^4-269746176x_2^
------------------------------------------------------------------------
4x_5^3+2891341824x_2^4x_5^2+23243677632x_2^4x_5+373714754427x_2^4+
------------------------------------------------------------------------
39846304512x_2^3x_5^2+640653864732x_2^3x_5+528482304x_2^2x_5^5-
------------------------------------------------------------------------
1416167424x_2^2x_5^4+37948861440x_2^2x_5^3+366087922704x_2^2x_5^2+
------------------------------------------------------------------------
301989888x_2x_5^6-809238528x_2x_5^5+8674025472x_2x_5^4+69731032896x_2x_5
------------------------------------------------------------------------
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^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
7 2 12 2
o13 = (map(R,R,{-x + x + x , x , -x + x + x , x }), ideal (--x + x x +
5 1 2 4 1 3 1 2 3 2 5 1 1 2
-----------------------------------------------------------------------
14 3 31 2 2 3 7 2 2 2 2 2
x x + 1, --x x + --x x + x x + -x x x + x x x + -x x x + x x x
1 4 15 1 2 15 1 2 1 2 5 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
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 5 8 1 17 2
o16 = (map(R,R,{--x + -x + x , x , -x + -x + x , x }), ideal (--x +
7 1 9 2 4 1 3 1 3 2 3 2 7 1
-----------------------------------------------------------------------
5 80 3 370 2 2 5 3 10 2 5 2
-x x + x x + 1, --x x + ---x x + --x x + --x x x + -x x x +
9 1 2 1 4 21 1 2 189 1 2 27 1 2 7 1 2 3 9 1 2 3
-----------------------------------------------------------------------
8 2 1 2
-x x x + -x x x + x x x x + 1), {x , x })
3 1 2 4 3 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,{2x + x + x , x , x + 2x + x , x }), ideal (3x + x x +
1 2 4 1 1 2 3 2 1 1 2
-----------------------------------------------------------------------
3 2 2 3 2 2 2 2
x x + 1, 2x x + 5x x + 2x x + 2x x x + x x x + x x x + 2x x x +
1 4 1 2 1 2 1 2 1 2 3 1 2 3 1 2 4 1 2 4
-----------------------------------------------------------------------
x x x x + 1), {x , x })
1 2 3 4 4 3
o19 : Sequence
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This symbol is provided by the package NoetherNormalization.