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
4 1 7 9 2
o3 = (map(R,R,{-x + x + x , x , -x + -x + x , x }), ideal (-x + x x +
5 1 2 4 1 3 1 5 2 3 2 5 1 1 2
------------------------------------------------------------------------
4 3 109 2 2 7 3 4 2 2 1 2
x x + 1, --x x + ---x x + -x x + -x x x + x x x + -x x x +
1 4 15 1 2 75 1 2 5 1 2 5 1 2 3 1 2 3 3 1 2 4
------------------------------------------------------------------------
7 2
-x x x + x x x x + 1), {x , x })
5 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 6 1
o6 = (map(R,R,{9x + -x + x , x , -x + -x + x , 4x + -x + x , x }),
1 9 2 5 1 4 1 7 2 4 1 2 2 3 2
------------------------------------------------------------------------
2 1 3 3 2 2 2 1 3
ideal (9x + -x x + x x - x , 729x x + 27x x + 243x x x + -x x +
1 9 1 2 1 5 2 1 2 1 2 1 2 5 3 1 2
------------------------------------------------------------------------
2 2 1 4 1 3 1 2 2 3
6x x x + 27x x x + ---x + --x x + -x x + x x ), {x , x , x })
1 2 5 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} | 59049x_1x_2x_5^6-39366x_2^9x_5-x_2^9+177147x_2^8x
{-9} | 81x_1x_2^2x_5^3-14348907x_1x_2x_5^5+729x_1x_2x_5^
{-9} | 81x_1x_2^3+14348907x_1x_2^2x_5^2+1458x_1x_2^2x_5+
{-3} | 81x_1^2+x_1x_2+9x_1x_5-9x_2^3
------------------------------------------------------------------------
_5^2+9x_2^8x_5-531441x_2^7x_5^3-81x_2^7x_5^2+729x_2^6x_5^3-6561x_2^5x_5^
4+9565938x_2^9-43046721x_2^8x_5-729x_2^8+129140163x_2^7x_5^2+13122x_2^7x
33354363399333138x_1x_2x_5^5-847288609443x_1x_2x_5^4+86093442x_1x_2x_5^3
------------------------------------------------------------------------
4+59049x_2^4x_5^5+729x_2^2x_5^6+6561x_2x_5^7
_5-177147x_2^6x_5^2+1594323x_2^5x_5^3-14348907x_2^4x_5^4+729x_2^4x_
+6561x_1x_2x_5^2-22236242266222092x_2^9+100063090197999414x_2^8x_5+
------------------------------------------------------------------------
5^3+x_2^3x_5^3-177147x_2^2x_5^5+18x_2^2x_5^4-1594323x_2x_5^6+81x_2x_5^5
2541865828329x_2^8-300189270593998242x_2^7x_5^2-38127987424935x_2^7x_5+
------------------------------------------------------------------------
387420489x_2^7+411782264189298x_2^6x_5^2-10460353203x_2^6x_5-531441x_2^6
------------------------------------------------------------------------
-3706040377703682x_2^5x_5^3+94143178827x_2^5x_5^2+4782969x_2^5x_5+729x_2
------------------------------------------------------------------------
^5+33354363399333138x_2^4x_5^4-847288609443x_2^4x_5^3+86093442x_2^4x_5^2
------------------------------------------------------------------------
+6561x_2^4x_5+x_2^4+177147x_2^3x_5^2+27x_2^3x_5+411782264189298x_2^2x_5^
------------------------------------------------------------------------
5-10460353203x_2^2x_5^4+2657205x_2^2x_5^3+243x_2^2x_5^2+
------------------------------------------------------------------------
3706040377703682x_2x_5^6-94143178827x_2x_5^5+9565938x_2x_5^4+729x_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
5 2 2 3 11 2 2
o13 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (--x + -x x
6 1 9 2 4 1 3 1 8 2 3 2 6 1 9 1 2
-----------------------------------------------------------------------
5 3 199 2 2 1 3 5 2 2 2 2 2
+ x x + 1, -x x + ---x x + --x x + -x x x + -x x x + -x x x +
1 4 9 1 2 432 1 2 12 1 2 6 1 2 3 9 1 2 3 3 1 2 4
-----------------------------------------------------------------------
3 2
-x x x + x x x x + 1), {x , x })
8 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
1 1 6 7 3 2 1
o16 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (-x + -x x
2 1 6 2 4 1 5 1 8 2 3 2 2 1 6 1 2
-----------------------------------------------------------------------
3 3 51 2 2 7 3 1 2 1 2 6 2
+ x x + 1, -x x + --x x + --x x + -x x x + -x x x + -x x x +
1 4 5 1 2 80 1 2 48 1 2 2 1 2 3 6 1 2 3 5 1 2 4
-----------------------------------------------------------------------
7 2
-x x x + x x x x + 1), {x , x })
8 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
--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: 5
--trying with basis element limit: 5
--trying with basis element limit: 20
--warning: no good linear transformation found by noetherNormalization
2
o19 = (map(R,R,{5x + 23x + x , x , 5x + x + x , x }), ideal (6x + 23x x
1 2 4 1 1 2 3 2 1 1 2
-----------------------------------------------------------------------
3 2 2 3 2 2 2
+ x x + 1, 25x x + 120x x + 23x x + 5x x x + 23x x x + 5x x x +
1 4 1 2 1 2 1 2 1 2 3 1 2 3 1 2 4
-----------------------------------------------------------------------
2
x x x + x x x x + 1), {x , x })
1 2 4 1 2 3 4 4 3
o19 : Sequence
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This symbol is provided by the package NoetherNormalization.