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 1 5 2 3
o3 = (map(R,R,{2x + -x + x , x , --x + -x + x , x }), ideal (3x + -x x
1 5 2 4 1 10 1 4 2 3 2 1 5 1 2
------------------------------------------------------------------------
1 3 64 2 2 3 3 2 3 2 1 2
+ x x + 1, -x x + --x x + -x x + 2x x x + -x x x + --x x x +
1 4 5 1 2 25 1 2 4 1 2 1 2 3 5 1 2 3 10 1 2 4
------------------------------------------------------------------------
5 2
-x x x + x x x x + 1), {x , x })
4 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
3 5 5 1
o6 = (map(R,R,{x + -x + x , x , -x + -x + x , -x + x + x , x }), ideal
1 7 2 5 1 8 1 7 2 4 5 1 2 3 2
------------------------------------------------------------------------
2 3 3 3 9 2 2 2 27 3 18 2
(x + -x x + x x - x , x x + -x x + 3x x x + --x x + --x x x +
1 7 1 2 1 5 2 1 2 7 1 2 1 2 5 49 1 2 7 1 2 5
------------------------------------------------------------------------
2 27 4 27 3 9 2 2 3
3x x x + ---x + --x x + -x x + x x ), {x , x , x })
1 2 5 343 2 49 2 5 7 2 5 2 5 5 4 3
o6 : Sequence
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i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 16807x_1x_2x_5^6-18522x_2^9x_5-243x_2^9+21609x_2^8x_5^2
{-9} | 189x_1x_2^2x_5^3-16807x_1x_2x_5^5+441x_1x_2x_5^4+18522x
{-9} | 137781x_1x_2^3+12252303x_1x_2^2x_5^2+642978x_1x_2^2x_5+
{-3} | 7x_1^2+3x_1x_2+7x_1x_5-7x_2^3
------------------------------------------------------------------------
+567x_2^8x_5-16807x_2^7x_5^3-1323x_2^7x_5^2+3087x_2^6x_5^3-7203x_2^5x_5
_2^9-21609x_2^8x_5-189x_2^8+16807x_2^7x_5^2+882x_2^7x_5-3087x_2^6x_5^2+
27682574402x_1x_2x_5^5-363182463x_1x_2x_5^4+19059138x_1x_2x_5^3+750141x
------------------------------------------------------------------------
^4+16807x_2^4x_5^5+7203x_2^2x_5^6+16807x_2x_5^7
7203x_2^5x_5^3-16807x_2^4x_5^4+441x_2^4x_5^3+81x_2^3x_5^3-7203x
_1x_2x_5^2-30507326892x_2^9+35591881374x_2^8x_5+466948881x_2^8-
------------------------------------------------------------------------
_2^2x_5^5+378x_2^2x_5^4-16807x_2x_5^6+441x_2x_5^5
27682574402x_2^7x_5^2-1815912315x_2^7x_5+9529569x_2^7+5084554482x_2^6x_5
------------------------------------------------------------------------
^2-66706983x_2^6x_5-1750329x_2^6-11863960458x_2^5x_5^3+155649627x_2^5x_5
------------------------------------------------------------------------
^2+4084101x_2^5x_5+321489x_2^5+27682574402x_2^4x_5^4-363182463x_2^4x_5^3
------------------------------------------------------------------------
+19059138x_2^4x_5^2+750141x_2^4x_5+59049x_2^4+5250987x_2^3x_5^2+413343x_
------------------------------------------------------------------------
2^3x_5+11863960458x_2^2x_5^5-155649627x_2^2x_5^4+20420505x_2^2x_5^3+
------------------------------------------------------------------------
964467x_2^2x_5^2+27682574402x_2x_5^6-363182463x_2x_5^5+19059138x_2x_5^4+
------------------------------------------------------------------------
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750141x_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
3 9 5 3 7 2
o13 = (map(R,R,{-x + --x + x , x , -x + --x + x , x }), ideal (-x +
4 1 10 2 4 1 4 1 10 2 3 2 4 1
-----------------------------------------------------------------------
9 15 3 27 2 2 27 3 3 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 16 1 2 20 1 2 100 1 2 4 1 2 3 10 1 2 3
-----------------------------------------------------------------------
5 2 3 2
-x x x + --x x x + x x x x + 1), {x , x })
4 1 2 4 10 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
3 9 2 3
o16 = (map(R,R,{3x + -x + x , x , 2x + -x + x , x }), ideal (4x + -x x
1 4 2 4 1 1 7 2 3 2 1 4 1 2
-----------------------------------------------------------------------
3 75 2 2 27 3 2 3 2 2
+ x x + 1, 6x x + --x x + --x x + 3x x x + -x x x + 2x x x +
1 4 1 2 14 1 2 28 1 2 1 2 3 4 1 2 3 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
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
2
o19 = (map(R,R,{2x - 3x + x , x , x + x , x }), ideal (3x - 3x x + x x
1 2 4 1 2 3 2 1 1 2 1 4
-----------------------------------------------------------------------
2 2 3 2 2 2
+ 1, 2x x - 3x x + 2x x x - 3x x x + x x x + x x x x + 1), {x ,
1 2 1 2 1 2 3 1 2 3 1 2 4 1 2 3 4 4
-----------------------------------------------------------------------
x })
3
o19 : Sequence
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This symbol is provided by the package NoetherNormalization.