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 7 3 2 5
o3 = (map(R,R,{4x + -x + x , x , -x + -x + x , x }), ideal (5x + -x x +
1 9 2 4 1 2 1 8 2 3 2 1 9 1 2
------------------------------------------------------------------------
3 31 2 2 5 3 2 5 2 7 2
x x + 1, 14x x + --x x + --x x + 4x x x + -x x x + -x x x +
1 4 1 2 9 1 2 24 1 2 1 2 3 9 1 2 3 2 1 2 4
------------------------------------------------------------------------
3 2
-x x x + x x x x + 1), {x , x })
8 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 10 2 4 3 3
o6 = (map(R,R,{--x + --x + x , x , -x + -x + x , -x + -x + x , x }),
10 1 7 2 5 1 3 1 7 2 4 8 1 4 2 3 2
------------------------------------------------------------------------
3 2 10 3 27 3 27 2 2 27 2 90 3
ideal (--x + --x x + x x - x , ----x x + --x x + ---x x x + --x x
10 1 7 1 2 1 5 2 1000 1 2 70 1 2 100 1 2 5 49 1 2
------------------------------------------------------------------------
18 2 9 2 1000 4 300 3 30 2 2 3
+ --x x x + --x x x + ----x + ---x x + --x x + x x ), {x , x , x })
7 1 2 5 10 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} | 504210x_1x_2x_5^6-1852200x_2^9x_5-3000000x_2^9+648270x_2
{-9} | 21000000x_1x_2^2x_5^3-4537890x_1x_2x_5^5+14700000x_1x_2x
{-9} | 10500000000000000x_1x_2^3+2268945000000000x_1x_2^2x_5^2+
{-3} | 21x_1^2+100x_1x_2+70x_1x_5-70x_2^3
------------------------------------------------------------------------
^8x_5^2+2100000x_2^8x_5-151263x_2^7x_5^3-1470000x_2^7x_5
_5^4+16669800x_2^9-5834430x_2^8x_5-6300000x_2^8+1361367x
14700000000000000x_1x_2^2x_5+100902983695290x_1x_2x_5^5-
------------------------------------------------------------------------
^2+1029000x_2^6x_5^3-720300x_2^5x_5^4+504210x_2^4x_5^5+2401000x_2^2x_5^6
_2^7x_5^2+8820000x_2^7x_5-9261000x_2^6x_5^2+6482700x_2^5x_5^3-4537890x_2
163432108350000x_1x_2x_5^4+1058841000000000x_1x_2x_5^3+5145000000000000x
------------------------------------------------------------------------
+1680700x_2x_5^7
^4x_5^4+14700000x_2^4x_5^3+100000000x_2^3x_5^3-21609000x
_1x_2x_5^2-370664021737800x_2^9+129732407608230x_2^8x_5+
------------------------------------------------------------------------
_2^2x_5^5+140000000x_2^2x_5^4-15126300x_2x_5^6+49000000x_2x_5^5
210126996450000x_2^8-30270895108587x_2^7x_5^2-245148162525000x_2^7x_5+
------------------------------------------------------------------------
158826150000000x_2^7+205924456521000x_2^6x_5^2-333534915000000x_2^6x_5-
------------------------------------------------------------------------
1080450000000000x_2^6-144147119564700x_2^5x_5^3+233474440500000x_2^5x_5^
------------------------------------------------------------------------
2+756315000000000x_2^5x_5+7350000000000000x_2^5+100902983695290x_2^4x_5^
------------------------------------------------------------------------
4-163432108350000x_2^4x_5^3+1058841000000000x_2^4x_5^2+5145000000000000x
------------------------------------------------------------------------
_2^4x_5+50000000000000000x_2^4+10804500000000000x_2^3x_5^2+
------------------------------------------------------------------------
105000000000000000x_2^3x_5+480490398549000x_2^2x_5^5-778248135000000x_2^
------------------------------------------------------------------------
2x_5^4+12605250000000000x_2^2x_5^3+73500000000000000x_2^2x_5^2+
------------------------------------------------------------------------
336343278984300x_2x_5^6-544773694500000x_2x_5^5+3529470000000000x_2x_5^4
------------------------------------------------------------------------
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+17150000000000000x_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
10 7 1 13 2
o13 = (map(R,R,{--x + 5x + x , x , -x + -x + x , x }), ideal (--x +
3 1 2 4 1 2 1 6 2 3 2 3 1
-----------------------------------------------------------------------
35 3 325 2 2 5 3 10 2 2
5x x + x x + 1, --x x + ---x x + -x x + --x x x + 5x x x +
1 2 1 4 3 1 2 18 1 2 6 1 2 3 1 2 3 1 2 3
-----------------------------------------------------------------------
7 2 1 2
-x x x + -x x x + x x x x + 1), {x , x })
2 1 2 4 6 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 8 1 3 5 2 8
o16 = (map(R,R,{-x + -x + x , x , -x + -x + x , x }), ideal (-x + -x x
2 1 9 2 4 1 6 1 7 2 3 2 2 1 9 1 2
-----------------------------------------------------------------------
1 3 299 2 2 8 3 3 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 4 1 2 378 1 2 21 1 2 2 1 2 3 9 1 2 3 6 1 2 4
-----------------------------------------------------------------------
3 2
-x x x + x x x x + 1), {x , x })
7 1 2 4 1 2 3 4 4 3
o16 : Sequence
|
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,{- 4x - x + x , x , - 4x + 2x + x , x }), ideal (- 3x -
1 2 4 1 1 2 3 2 1
-----------------------------------------------------------------------
3 2 2 3 2 2 2
x x + x x + 1, 16x x - 4x x - 2x x - 4x x x - x x x - 4x x x +
1 2 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.