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
5 1 14 2
o3 = (map(R,R,{-x + 2x + x , x , -x + x + x , x }), ideal (--x + 2x x +
9 1 2 4 1 6 1 2 3 2 9 1 1 2
------------------------------------------------------------------------
5 3 8 2 2 3 5 2 2 1 2 2
x x + 1, --x x + -x x + 2x x + -x x x + 2x x x + -x x x + x x x
1 4 54 1 2 9 1 2 1 2 9 1 2 3 1 2 3 6 1 2 4 1 2 4
------------------------------------------------------------------------
+ x x x x + 1), {x , x })
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 8 4 2
o6 = (map(R,R,{-x + -x + x , x , x + -x + x , -x + -x + x , x }), ideal
5 1 7 2 5 1 1 3 2 4 7 1 3 2 3 2
------------------------------------------------------------------------
2 2 4 3 8 3 48 2 2 12 2 96 3
(-x + -x x + x x - x , ---x x + ---x x + --x x x + ---x x +
5 1 7 1 2 1 5 2 125 1 2 175 1 2 25 1 2 5 245 1 2
------------------------------------------------------------------------
48 2 6 2 64 4 48 3 12 2 2 3
--x x x + -x x x + ---x + --x x + --x x + x x ), {x , x , x })
35 1 2 5 5 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} | 168070x_1x_2x_5^6-131712x_2^9x_5-10240x_2^9+115248x_2^8x_5^2+
{-9} | 44800x_1x_2^2x_5^3-504210x_1x_2x_5^5+78400x_1x_2x_5^4+395136x
{-9} | 2293760000x_1x_2^3+25815552000x_1x_2^2x_5^2+8028160000x_1x_2^
{-3} | 14x_1^2+20x_1x_2+35x_1x_5-35x_2^3
------------------------------------------------------------------------
17920x_2^8x_5-67228x_2^7x_5^3-31360x_2^7x_5^2+54880x_2^6x_5^3-96040x_2^
_2^9-345744x_2^8x_5-17920x_2^8+201684x_2^7x_5^2+62720x_2^7x_5-164640x_2
2x_5+1245715848090x_1x_2x_5^5-96848656800x_1x_2x_5^4+30118144000x_1x_2x
------------------------------------------------------------------------
5x_5^4+168070x_2^4x_5^5+240100x_2^2x_5^6+420175x_2x_5^7
^6x_5^2+288120x_2^5x_5^3-504210x_2^4x_5^4+78400x_2^4x_5^3+64000x_2
_5^3+7024640000x_1x_2x_5^2-976234460544x_2^9+854205152976x_2^8x_5+
------------------------------------------------------------------------
^3x_5^3-720300x_2^2x_5^5+224000x_2^2x_5^4-1260525x_2x_5^6+196000x_2x_5^5
66410507520x_2^8-498286339236x_2^7x_5^2-193697313600x_2^7x_5+6023628800x
------------------------------------------------------------------------
_2^7+406764358560x_2^6x_5^2-31624051200x_2^6x_5-4917248000x_2^6-
------------------------------------------------------------------------
711837627480x_2^5x_5^3+55342089600x_2^5x_5^2+8605184000x_2^5x_5+
------------------------------------------------------------------------
4014080000x_2^5+1245715848090x_2^4x_5^4-96848656800x_2^4x_5^3+
------------------------------------------------------------------------
30118144000x_2^4x_5^2+7024640000x_2^4x_5+3276800000x_2^4+36879360000x_2^
------------------------------------------------------------------------
3x_5^2+17203200000x_2^3x_5+1779594068700x_2^2x_5^5-138355224000x_2^2x_5^
------------------------------------------------------------------------
4+107564800000x_2^2x_5^3+30105600000x_2^2x_5^2+3114289620225x_2x_5^6-
------------------------------------------------------------------------
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242121642000x_2x_5^5+75295360000x_2x_5^4+17561600000x_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
8 8 1 17 2 8
o13 = (map(R,R,{-x + -x + x , x , -x + x + x , x }), ideal (--x + -x x
9 1 3 2 4 1 3 1 2 3 2 9 1 3 1 2
-----------------------------------------------------------------------
8 3 16 2 2 8 3 8 2 8 2 1 2
+ x x + 1, --x x + --x x + -x x + -x x x + -x x x + -x x x +
1 4 27 1 2 9 1 2 3 1 2 9 1 2 3 3 1 2 3 3 1 2 4
-----------------------------------------------------------------------
2
x x x + x x x x + 1), {x , x })
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 13 2
o16 = (map(R,R,{--x + 4x + x , x , --x + -x + x , x }), ideal (--x +
10 1 2 4 1 10 1 3 2 3 2 10 1
-----------------------------------------------------------------------
27 3 19 2 2 8 3 3 2 2
4x x + x x + 1, ---x x + --x x + -x x + --x x x + 4x x x +
1 2 1 4 100 1 2 5 1 2 3 1 2 10 1 2 3 1 2 3
-----------------------------------------------------------------------
9 2 2 2
--x x x + -x x x + x x x x + 1), {x , x })
10 1 2 4 3 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 - x + x , x }), ideal (x + 2x x + x x +
2 4 1 1 2 3 2 1 1 2 1 4
-----------------------------------------------------------------------
2 2 3 2 2 2
1, 2x x - 2x x + 2x x x + x x x - x x x + x x x x + 1), {x , x })
1 2 1 2 1 2 3 1 2 4 1 2 4 1 2 3 4 4 3
o19 : Sequence
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This symbol is provided by the package NoetherNormalization.