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
9 1 17 2
o3 = (map(R,R,{-x + x + x , x , -x + 10x + x , x }), ideal (--x + x x +
8 1 2 4 1 2 1 2 3 2 8 1 1 2
------------------------------------------------------------------------
9 3 47 2 2 3 9 2 2 1 2
x x + 1, --x x + --x x + 10x x + -x x x + x x x + -x x x +
1 4 16 1 2 4 1 2 1 2 8 1 2 3 1 2 3 2 1 2 4
------------------------------------------------------------------------
2
10x 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
2 4 1 9 2 7
o6 = (map(R,R,{-x + -x + x , x , -x + --x + x , -x + -x + x , x }),
3 1 3 2 5 1 8 1 10 2 4 9 1 4 2 3 2
------------------------------------------------------------------------
2 2 4 3 8 3 16 2 2 4 2 32 3
ideal (-x + -x x + x x - x , --x x + --x x + -x x x + --x x +
3 1 3 1 2 1 5 2 27 1 2 9 1 2 3 1 2 5 9 1 2
------------------------------------------------------------------------
16 2 2 64 4 16 3 2 2 3
--x x x + 2x x x + --x + --x x + 4x x + x x ), {x , x , x })
3 1 2 5 1 2 5 27 2 3 2 5 2 5 2 5 5 4 3
o6 : Sequence
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i7 : transpose gens gb J
o7 = {-10} | x_2^10
{-10} | 486x_1x_2x_5^6-3456x_2^9x_5-2048x_2^9+1296x_2^8x_5^2+1536x_2^8x_
{-9} | 768x_1x_2^2x_5^3-486x_1x_2x_5^5+576x_1x_2x_5^4+3456x_2^9-1296x_2
{-9} | 524288x_1x_2^3+331776x_1x_2^2x_5^2+786432x_1x_2^2x_5+118098x_1x_
{-3} | 2x_1^2+4x_1x_2+3x_1x_5-3x_2^3
------------------------------------------------------------------------
5-324x_2^7x_5^3-1152x_2^7x_5^2+864x_2^6x_5^3-648x_2^5x_5^4+486x_2^4x_
^8x_5-512x_2^8+324x_2^7x_5^2+768x_2^7x_5-864x_2^6x_5^2+648x_2^5x_5^3-
2x_5^5-69984x_1x_2x_5^4+165888x_1x_2x_5^3+294912x_1x_2x_5^2-839808x_2
------------------------------------------------------------------------
5^5+972x_2^2x_5^6+729x_2x_5^7
486x_2^4x_5^4+576x_2^4x_5^3+1536x_2^3x_5^3-972x_2^2x_5^5+2304x_2^2x_5^4-
^9+314928x_2^8x_5+186624x_2^8-78732x_2^7x_5^2-233280x_2^7x_5+55296x_2^7+
------------------------------------------------------------------------
729x_2x_5^6+864x_2x_5^5
209952x_2^6x_5^2-124416x_2^6x_5-147456x_2^6-157464x_2^5x_5^3+93312x_2^5x
------------------------------------------------------------------------
_5^2+110592x_2^5x_5+393216x_2^5+118098x_2^4x_5^4-69984x_2^4x_5^3+165888x
------------------------------------------------------------------------
_2^4x_5^2+294912x_2^4x_5+1048576x_2^4+663552x_2^3x_5^2+2359296x_2^3x_5+
------------------------------------------------------------------------
236196x_2^2x_5^5-139968x_2^2x_5^4+829440x_2^2x_5^3+1769472x_2^2x_5^2+
------------------------------------------------------------------------
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177147x_2x_5^6-104976x_2x_5^5+248832x_2x_5^4+442368x_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 5 2 2 4
o13 = (map(R,R,{3x + -x + x , x , -x + -x + x , x }), ideal (4x + -x x
1 3 2 4 1 2 1 3 2 3 2 1 3 1 2
-----------------------------------------------------------------------
15 3 16 2 2 8 3 2 4 2 5 2
+ x x + 1, --x x + --x x + -x x + 3x x x + -x x x + -x x x +
1 4 2 1 2 3 1 2 9 1 2 1 2 3 3 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
4 2
o16 = (map(R,R,{x + 6x + x , x , 8x + -x + x , x }), ideal (2x + 6x x +
1 2 4 1 1 7 2 3 2 1 1 2
-----------------------------------------------------------------------
3 340 2 2 24 3 2 2 2
x x + 1, 8x x + ---x x + --x x + x x x + 6x x x + 8x x x +
1 4 1 2 7 1 2 7 1 2 1 2 3 1 2 3 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
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 - x + x , x , - 2x - 2x + x , x }), ideal (3x - x x
1 2 4 1 1 2 3 2 1 1 2
-----------------------------------------------------------------------
3 2 2 3 2 2 2
+ x x + 1, - 4x x - 2x x + 2x x + 2x x x - x x x - 2x x x -
1 4 1 2 1 2 1 2 1 2 3 1 2 3 1 2 4
-----------------------------------------------------------------------
2
2x 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.