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 1 3 4 14 2 1
o3 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (--x + -x x
9 1 3 2 4 1 2 1 7 2 3 2 9 1 3 1 2
------------------------------------------------------------------------
5 3 103 2 2 4 3 5 2 1 2 3 2
+ x x + 1, -x x + ---x x + --x x + -x x x + -x x x + -x x x +
1 4 6 1 2 126 1 2 21 1 2 9 1 2 3 3 1 2 3 2 1 2 4
------------------------------------------------------------------------
4 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
8 10 1 4 1 8
o6 = (map(R,R,{-x + --x + x , x , -x + -x + x , -x + -x + x , x }),
9 1 9 2 5 1 2 1 9 2 4 2 1 9 2 3 2
------------------------------------------------------------------------
8 2 10 3 512 3 640 2 2 64 2 800 3
ideal (-x + --x x + x x - x , ---x x + ---x x + --x x x + ---x x
9 1 9 1 2 1 5 2 729 1 2 243 1 2 27 1 2 5 243 1 2
------------------------------------------------------------------------
160 2 8 2 1000 4 100 3 10 2 2 3
+ ---x x x + -x x x + ----x + ---x x + --x x + x x ), {x , x , x })
27 1 2 5 3 1 2 5 729 2 27 2 5 3 2 5 2 5 5 4 3
o6 : Sequence
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i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 472392x_1x_2x_5^6-3110400x_2^9x_5-800000x_2^9+1399680x_2^8x_5^2+
{-9} | 90000x_1x_2^2x_5^3-157464x_1x_2x_5^5+81000x_1x_2x_5^4+1036800x_2
{-9} | 1562500000x_1x_2^3+2733750000x_1x_2^2x_5^2+2812500000x_1x_2^2x_5
{-3} | 8x_1^2+10x_1x_2+9x_1x_5-9x_2^3
------------------------------------------------------------------------
720000x_2^8x_5-419904x_2^7x_5^3-648000x_2^7x_5^2+583200x_2^
^9-466560x_2^8x_5-80000x_2^8+139968x_2^7x_5^2+144000x_2^7x_
+6198727824x_1x_2x_5^5-1594323000x_1x_2x_5^4+1640250000x_1x
------------------------------------------------------------------------
6x_5^3-524880x_2^5x_5^4+472392x_2^4x_5^5+590490x_2^2x_5^6+531441x_
5-194400x_2^6x_5^2+174960x_2^5x_5^3-157464x_2^4x_5^4+81000x_2^4x_5
_2x_5^3+1265625000x_1x_2x_5^2-40814668800x_2^9+18366600960x_2^8x_5
------------------------------------------------------------------------
2x_5^7
^3+112500x_2^3x_5^3-196830x_2^2x_5^5+202500x_2^2x_5^4-177147x_2x_5^6+
+4723920000x_2^8-5509980288x_2^7x_5^2-7085880000x_2^7x_5+729000000x_2
------------------------------------------------------------------------
91125x_2x_5^5
^7+7652750400x_2^6x_5^2-1968300000x_2^6x_5-1012500000x_2^6-6887475360x_2
------------------------------------------------------------------------
^5x_5^3+1771470000x_2^5x_5^2+911250000x_2^5x_5+1406250000x_2^5+
------------------------------------------------------------------------
6198727824x_2^4x_5^4-1594323000x_2^4x_5^3+1640250000x_2^4x_5^2+
------------------------------------------------------------------------
1265625000x_2^4x_5+1953125000x_2^4+3417187500x_2^3x_5^2+5273437500x_2^3x
------------------------------------------------------------------------
_5+7748409780x_2^2x_5^5-1992903750x_2^2x_5^4+5125781250x_2^2x_5^3+
------------------------------------------------------------------------
4746093750x_2^2x_5^2+6973568802x_2x_5^6-1793613375x_2x_5^5+1845281250x_
------------------------------------------------------------------------
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2x_5^4+1423828125x_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
1 5 2 2 1
o13 = (map(R,R,{2x + -x + x , x , -x + -x + x , x }), ideal (3x + -x x
1 6 2 4 1 2 1 3 2 3 2 1 6 1 2
-----------------------------------------------------------------------
3 7 2 2 1 3 2 1 2 5 2
+ x x + 1, 5x x + -x x + -x x + 2x x x + -x x x + -x x x +
1 4 1 2 4 1 2 9 1 2 1 2 3 6 1 2 3 2 1 2 4
-----------------------------------------------------------------------
2 2
-x x x + x x x x + 1), {x , x })
3 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
5 9 1 4 7 2
o16 = (map(R,R,{-x + --x + x , x , -x + -x + x , x }), ideal (-x +
2 1 10 2 4 1 9 1 3 2 3 2 2 1
-----------------------------------------------------------------------
9 5 3 103 2 2 6 3 5 2 9 2
--x x + x x + 1, --x x + ---x x + -x x + -x x x + --x x x +
10 1 2 1 4 18 1 2 30 1 2 5 1 2 2 1 2 3 10 1 2 3
-----------------------------------------------------------------------
1 2 4 2
-x x x + -x x x + x x x x + 1), {x , x })
9 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
--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
--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: 4
--trying with basis element limit: 5
--trying with basis element limit: 20
2
o19 = (map(R,R,{4x - x + x , x , - 4x - x + x , x }), ideal (5x - x x +
1 2 4 1 1 2 3 2 1 1 2
-----------------------------------------------------------------------
3 3 2 2 2 2
x x + 1, - 16x x + x x + 4x x x - x x x - 4x x x - x x x +
1 4 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.