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
2 5 7 9 7 2 5
o3 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (-x + -x x +
5 1 4 2 4 1 6 1 2 2 3 2 5 1 4 1 2
------------------------------------------------------------------------
7 3 391 2 2 45 3 2 2 5 2 7 2
x x + 1, --x x + ---x x + --x x + -x x x + -x x x + -x x x +
1 4 15 1 2 120 1 2 8 1 2 5 1 2 3 4 1 2 3 6 1 2 4
------------------------------------------------------------------------
9 2
-x x x + x x x x + 1), {x , x })
2 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 10 6 7 3
o6 = (map(R,R,{-x + -x + x , x , --x + -x + x , -x + -x + x , x }),
4 1 6 2 5 1 3 1 5 2 4 4 1 4 2 3 2
------------------------------------------------------------------------
1 2 7 3 1 3 7 2 2 3 2 49 3
ideal (-x + -x x + x x - x , --x x + --x x + --x x x + --x x +
4 1 6 1 2 1 5 2 64 1 2 32 1 2 16 1 2 5 48 1 2
------------------------------------------------------------------------
7 2 3 2 343 4 49 3 7 2 2 3
-x x x + -x x x + ---x + --x x + -x x + x x ), {x , x , x })
4 1 2 5 4 1 2 5 216 2 12 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} | 7776x_1x_2x_5^6-15876x_2^9x_5-16807x_2^9+6804x_2^8x_5^2+14406x_2
{-9} | 28812x_1x_2^2x_5^3-11664x_1x_2x_5^5+24696x_1x_2x_5^4+23814x_2^9-
{-9} | 5931980229x_1x_2^3+2401451388x_1x_2^2x_5^2+10169108964x_1x_2^2x_
{-3} | 3x_1^2+14x_1x_2+12x_1x_5-12x_2^3
------------------------------------------------------------------------
^8x_5-1944x_2^7x_5^3-12348x_2^7x_5^2+10584x_2^6x_5^3-9072x_2^5x_5^
10206x_2^8x_5-7203x_2^8+2916x_2^7x_5^2+12348x_2^7x_5-15876x_2^6x_5
5+306110016x_1x_2x_5^5-324060912x_1x_2x_5^4+1372257936x_1x_2x_5^3+
------------------------------------------------------------------------
4+7776x_2^4x_5^5+36288x_2^2x_5^6+31104x_2x_5^7
^2+13608x_2^5x_5^3-11664x_2^4x_5^4+24696x_2^4x_5^3+134456x_2^3x_5^3-
4358189556x_1x_2x_5^2-624974616x_2^9+267846264x_2^8x_5+283553298x_2^
------------------------------------------------------------------------
54432x_2^2x_5^5+230496x_2^2x_5^4-46656x_2x_5^6+98784x_2x_5^5
8-76527504x_2^7x_5^2-405076140x_2^7x_5+171532242x_2^7+416649744x_2^6x_5^
------------------------------------------------------------------------
2-441082908x_2^6x_5-933897762x_2^6-357128352x_2^5x_5^3+378071064x_2^5x_5
------------------------------------------------------------------------
^2+800483796x_2^5x_5+5084554482x_2^5+306110016x_2^4x_5^4-324060912x_2^4x
------------------------------------------------------------------------
_5^3+1372257936x_2^4x_5^2+4358189556x_2^4x_5+27682574402x_2^4+
------------------------------------------------------------------------
11206773144x_2^3x_5^2+71183762748x_2^3x_5+1428513408x_2^2x_5^5-
------------------------------------------------------------------------
1512284256x_2^2x_5^4+16009675920x_2^2x_5^3+61014653784x_2^2x_5^2+
------------------------------------------------------------------------
1224440064x_2x_5^6-1296243648x_2x_5^5+5489031744x_2x_5^4+17432758224x_2x
------------------------------------------------------------------------
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_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
9 3 6 17 2
o13 = (map(R,R,{-x + 8x + x , x , -x + -x + x , x }), ideal (--x + 8x x
8 1 2 4 1 4 1 7 2 3 2 8 1 1 2
-----------------------------------------------------------------------
27 3 195 2 2 48 3 9 2 2 3 2
+ x x + 1, --x x + ---x x + --x x + -x x x + 8x x x + -x x x +
1 4 32 1 2 28 1 2 7 1 2 8 1 2 3 1 2 3 4 1 2 4
-----------------------------------------------------------------------
6 2
-x x x + x x x x + 1), {x , x })
7 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
4 8 9 1 7 2 8
o16 = (map(R,R,{-x + -x + x , x , --x + -x + x , x }), ideal (-x + -x x
3 1 7 2 4 1 10 1 2 2 3 2 3 1 7 1 2
-----------------------------------------------------------------------
6 3 178 2 2 4 3 4 2 8 2 9 2
+ x x + 1, -x x + ---x x + -x x + -x x x + -x x x + --x x x +
1 4 5 1 2 105 1 2 7 1 2 3 1 2 3 7 1 2 3 10 1 2 4
-----------------------------------------------------------------------
1 2
-x x x + x x x x + 1), {x , x })
2 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 - 2x + x , x , 2x + x + x , x }), ideal (2x - 2x 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 - 3x x - 2x x + x x x - 2x x x + 2x x x + x 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.