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
3 8 1 5 2 8
o3 = (map(R,R,{-x + -x + x , x , -x + 9x + x , x }), ideal (-x + -x x +
2 1 9 2 4 1 5 1 2 3 2 2 1 9 1 2
------------------------------------------------------------------------
3 3 1231 2 2 3 3 2 8 2 1 2
x x + 1, --x x + ----x x + 8x x + -x x x + -x x x + -x x x +
1 4 10 1 2 90 1 2 1 2 2 1 2 3 9 1 2 3 5 1 2 4
------------------------------------------------------------------------
2
9x x x + x x x x + 1), {x , x })
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
5 1 10 1
o6 = (map(R,R,{-x + -x + x , x , --x + x + x , x + -x + x , x }), ideal
3 1 9 2 5 1 7 1 2 4 1 7 2 3 2
------------------------------------------------------------------------
5 2 1 3 125 3 25 2 2 25 2 5 3
(-x + -x x + x x - x , ---x x + --x x + --x x x + --x x +
3 1 9 1 2 1 5 2 27 1 2 27 1 2 3 1 2 5 81 1 2
------------------------------------------------------------------------
10 2 2 1 4 1 3 1 2 2 3
--x x x + 5x x x + ---x + --x x + -x x + x x ), {x , x , x })
9 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} | 295245x_1x_2x_5^6-36450x_2^9x_5-5x_2^9+164025x
{-9} | 15x_1x_2^2x_5^3-492075x_1x_2x_5^5+135x_1x_2x_5
{-9} | 15x_1x_2^3+492075x_1x_2^2x_5^2+270x_1x_2^2x_5+
{-3} | 15x_1^2+x_1x_2+9x_1x_5-9x_2^3
------------------------------------------------------------------------
_2^8x_5^2+45x_2^8x_5-492075x_2^7x_5^3-405x_2^7x_5^2+3645x_2^6x_5^3
^4+60750x_2^9-273375x_2^8x_5-25x_2^8+820125x_2^7x_5^2+450x_2^7x_5-
39226324511250x_1x_2x_5^5-5380840125x_1x_2x_5^4+2952450x_1x_2x_5^3
------------------------------------------------------------------------
-32805x_2^5x_5^4+295245x_2^4x_5^5+19683x_2^2x_5^6+177147x_2x_5^7
6075x_2^6x_5^2+54675x_2^5x_5^3-492075x_2^4x_5^4+135x_2^4x_5^3+x_2^3x_5^3
+1215x_1x_2x_5^2-4842756112500x_2^9+21792402506250x_2^8x_5+2989355625x_2
------------------------------------------------------------------------
-32805x_2^2x_5^5+18x_2^2x_5^4-295245x_2x_5^6+81x_2x_5^5
^8-65377207518750x_2^7x_5^2-44840334375x_2^7x_5+2460375x_2^7+
------------------------------------------------------------------------
484275611250x_2^6x_5^2-66430125x_2^6x_5-18225x_2^6-4358480501250x_2^5x_5
------------------------------------------------------------------------
^3+597871125x_2^5x_5^2+164025x_2^5x_5+135x_2^5+39226324511250x_2^4x_5^4-
------------------------------------------------------------------------
5380840125x_2^4x_5^3+2952450x_2^4x_5^2+1215x_2^4x_5+x_2^4+32805x_2^3x_5^
------------------------------------------------------------------------
2+27x_2^3x_5+2615088300750x_2^2x_5^5-358722675x_2^2x_5^4+492075x_2^2x_5^
------------------------------------------------------------------------
3+243x_2^2x_5^2+23535794706750x_2x_5^6-3228504075x_2x_5^5+1771470x_2x_5^
------------------------------------------------------------------------
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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
4 8 4 9 7 2 8
o13 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (-x + -x x
3 1 5 2 4 1 5 1 7 2 3 2 3 1 5 1 2
-----------------------------------------------------------------------
16 3 524 2 2 72 3 4 2 8 2 4 2
+ x x + 1, --x x + ---x x + --x x + -x x x + -x x x + -x x x +
1 4 15 1 2 175 1 2 35 1 2 3 1 2 3 5 1 2 3 5 1 2 4
-----------------------------------------------------------------------
9 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
5 4 6 12 2 4
o16 = (map(R,R,{-x + -x + x , x , x + -x + x , x }), ideal (--x + -x x
7 1 9 2 4 1 1 5 2 3 2 7 1 9 1 2
-----------------------------------------------------------------------
5 3 82 2 2 8 3 5 2 4 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 63 1 2 15 1 2 7 1 2 3 9 1 2 3 1 2 4
-----------------------------------------------------------------------
6 2
-x x x + x x x x + 1), {x , x })
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
2
o19 = (map(R,R,{- 2x - 2x + x , x , - 2x + 2x + x , x }), ideal (- x -
1 2 4 1 1 2 3 2 1
-----------------------------------------------------------------------
3 3 2 2 2 2
2x x + x x + 1, 4x x - 4x x - 2x x x - 2x x x - 2x x x + 2x x x
1 2 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.