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noetherNormalization -- data for Noether normalization

Synopsis

Description

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];
i2 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);

o2 : Ideal of R
i3 : (f,J,X) = noetherNormalization I

               6     6             9     8                      11 2   6    
o3 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (--x  + -x x 
               5 1   7 2    4   1  7 1   7 2    3   2            5 1   7 1 2
     ------------------------------------------------------------------------
                 54 3     606 2 2   48   3   6 2       6   2     9 2      
     + x x  + 1, --x x  + ---x x  + --x x  + -x x x  + -x x x  + -x x x  +
        1 4      35 1 2   245 1 2   49 1 2   5 1 2 3   7 1 2 3   7 1 2 4  
     ------------------------------------------------------------------------
     8   2
     -x x x  + x x x x  + 1), {x , x })
     7 1 2 4    1 2 3 4         4   3

o3 : Sequence
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];
i5 : I = ideal(x_2*x_1-x_5^3, x_5*x_1^3);

o5 : Ideal of R
i6 : (f,J,X) = noetherNormalization I

               9     1                  1         7     7                    
o6 = (map(R,R,{-x  + -x  + x , x , x  + -x  + x , -x  + -x  + x , x }), ideal
               7 1   2 2    5   1   1   2 2    4  9 1   9 2    3   2         
     ------------------------------------------------------------------------
      9 2   1               3  729 3     243 2 2   243 2       27   3  
     (-x  + -x x  + x x  - x , ---x x  + ---x x  + ---x x x  + --x x  +
      7 1   2 1 2    1 5    2  343 1 2    98 1 2    49 1 2 5   28 1 2  
     ------------------------------------------------------------------------
     27   2     27     2   1 4   3 3     3 2 2      3
     --x x x  + --x x x  + -x  + -x x  + -x x  + x x ), {x , x , x })
      7 1 2 5    7 1 2 5   8 2   4 2 5   2 2 5    2 5     5   4   3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10                                            
     {-10} | 2016x_1x_2x_5^6-3888x_2^9x_5-63x_2^9+3888x_2^8x_5^
     {-9}  | 882x_1x_2^2x_5^3-54432x_1x_2x_5^5+1764x_1x_2x_5^4+
     {-9}  | 43218x_1x_2^3+2667168x_1x_2^2x_5^2+172872x_1x_2^2x
     {-3}  | 18x_1^2+7x_1x_2+14x_1x_5-14x_2^3                  
     ------------------------------------------------------------------------
                                                                             
     2+126x_2^8x_5-2592x_2^7x_5^3-252x_2^7x_5^2+504x_2^6x_5^3-1008x_2^5x_5^4+
     104976x_2^9-104976x_2^8x_5-1134x_2^8+69984x_2^7x_5^2+4536x_2^7x_5-13608x
     _5+3386105856x_1x_2x_5^5-54867456x_1x_2x_5^4+3556224x_1x_2x_5^3+172872x_
                                                                             
     ------------------------------------------------------------------------
                                                                            
     2016x_2^4x_5^5+784x_2^2x_5^6+1568x_2x_5^7                              
     _2^6x_5^2+27216x_2^5x_5^3-54432x_2^4x_5^4+1764x_2^4x_5^3+343x_2^3x_5^3-
     1x_2x_5^2-6530347008x_2^9+6530347008x_2^8x_5+105815808x_2^8-4353564672x
                                                                            
     ------------------------------------------------------------------------
                                                                             
                                                                             
     21168x_2^2x_5^5+1372x_2^2x_5^4-42336x_2x_5^6+1372x_2x_5^5               
     _2^7x_5^2-352719360x_2^7x_5+2286144x_2^7+846526464x_2^6x_5^2-13716864x_2
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     ^6x_5-444528x_2^6-1693052928x_2^5x_5^3+27433728x_2^5x_5^2+889056x_2^5x_5
                                                                             
     ------------------------------------------------------------------------
                                                                           
                                                                           
                                                                           
     +86436x_2^5+3386105856x_2^4x_5^4-54867456x_2^4x_5^3+3556224x_2^4x_5^2+
                                                                           
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     172872x_2^4x_5+16807x_2^4+1037232x_2^3x_5^2+100842x_2^3x_5+1316818944x_2
                                                                             
     ------------------------------------------------------------------------
                                                                   
                                                                   
                                                                   
     ^2x_5^5-21337344x_2^2x_5^4+3457440x_2^2x_5^3+201684x_2^2x_5^2+
                                                                   
     ------------------------------------------------------------------------
                                                                        |
                                                                        |
                                                                        |
     2633637888x_2x_5^6-42674688x_2x_5^5+2765952x_2x_5^4+134456x_2x_5^3 |
                                                                        |

             5       1
o7 : Matrix R  <--- R
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];
i9 : I = ideal(a^2*b+a*b^2+1);

o9 : Ideal of R
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
Here is an example with the option Verbose => true:
i11 : R = QQ[x_1..x_4];
i12 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);

o12 : Ideal of R
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                   2                            11 2        
o13 = (map(R,R,{-x  + 5x  + x , x , -x  + 4x  + x , x }), ideal (--x  + 5x x 
                7 1     2    4   1  5 1     2    3   2            7 1     1 2
      -----------------------------------------------------------------------
                   8 3     30 2 2        3   4 2           2     2 2      
      + x x  + 1, --x x  + --x x  + 20x x  + -x x x  + 5x x x  + -x x x  +
         1 4      35 1 2    7 1 2      1 2   7 1 2 3     1 2 3   5 1 2 4  
      -----------------------------------------------------------------------
          2
      4x x x  + x x x x  + 1), {x , x })
        1 2 4    1 2 3 4         4   3

o13 : Sequence
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]; 
i15 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);

o15 : Ideal of R
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

                10     7             7     2                      13 2  
o16 = (map(R,R,{--x  + -x  + x , x , -x  + -x  + x , x }), ideal (--x  +
                 3 1   2 2    4   1  4 1   5 2    3   2            3 1  
      -----------------------------------------------------------------------
      7                 35 3     179 2 2   7   3   10 2       7   2    
      -x x  + x x  + 1, --x x  + ---x x  + -x x  + --x x x  + -x x x  +
      2 1 2    1 4       6 1 2    24 1 2   5 1 2    3 1 2 3   2 1 2 3  
      -----------------------------------------------------------------------
      7 2       2   2
      -x x x  + -x x x  + x x x x  + 1), {x , x })
      4 1 2 4   5 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];
i18 : I = ideal(x_2^2+x_1*x_2+1, x_1*x_2*x_3*x_4+1);

o18 : Ideal of R
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  + 6x  + x , x , - 4x  - 2x  + x , x }), ideal (3x  +
                  1     2    4   1      1     2    3   2             1  
      -----------------------------------------------------------------------
                            3        2 2        3     2           2    
      6x x  + x x  + 1, - 8x x  - 28x x  - 12x x  + 2x x x  + 6x x x  -
        1 2    1 4          1 2      1 2      1 2     1 2 3     1 2 3  
      -----------------------------------------------------------------------
        2           2
      4x x x  - 2x x x  + x x x x  + 1), {x , x })
        1 2 4     1 2 4    1 2 3 4         4   3

o19 : Sequence

This symbol is provided by the package NoetherNormalization.

Ways to use noetherNormalization :