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NoetherNormalization :: noetherNormalization

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

               1                  3                            8 2         
o3 = (map(R,R,{-x  + x  + x , x , -x  + 2x  + x , x }), ideal (-x  + x x  +
               7 1    2    4   1  5 1     2    3   2           7 1    1 2  
     ------------------------------------------------------------------------
                3 3     31 2 2       3   1 2          2     3 2           2
     x x  + 1, --x x  + --x x  + 2x x  + -x x x  + x x x  + -x x x  + 2x x x 
      1 4      35 1 2   35 1 2     1 2   7 1 2 3    1 2 3   5 1 2 4     1 2 4
     ------------------------------------------------------------------------
     + x x x x  + 1), {x , x })
        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

               4     7                           5     1                    
o6 = (map(R,R,{-x  + -x  + x , x , x  + x  + x , -x  + -x  + x , x }), ideal
               7 1   9 2    5   1   1    2    4  2 1   2 2    3   2         
     ------------------------------------------------------------------------
      4 2   7               3   64 3     16 2 2   48 2       28   3   8   2  
     (-x  + -x x  + x x  - x , ---x x  + --x x  + --x x x  + --x x  + -x x x 
      7 1   9 1 2    1 5    2  343 1 2   21 1 2   49 1 2 5   27 1 2   3 1 2 5
     ------------------------------------------------------------------------
       12     2   343 4   49 3     7 2 2      3
     + --x x x  + ---x  + --x x  + -x x  + x x ), {x , x , x })
        7 1 2 5   729 2   27 2 5   3 2 5    2 5     5   4   3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10           
     {-10} | 1653372x_1x_2x_5^
     {-9}  | 4235364x_1x_2^2x_
     {-9}  | 170912214357948x_
     {-3}  | 36x_1^2+49x_1x_2+
     ------------------------------------------------------------------------
                                                              
     6-3429216x_2^9x_5-470596x_2^9+2204496x_2^8x_5^2+605052x_2
     5^3-19840464x_1x_2x_5^5+5445468x_1x_2x_5^4+41150592x_2^9-
     1x_2^3+800634286953648x_1x_2^2x_5^2+439488551206152x_1x_2
     63x_1x_5-63x_2^3                                         
     ------------------------------------------------------------------------
                                                               
     ^8x_5-944784x_2^7x_5^3-777924x_2^7x_5^2+1000188x_2^6x_5^3-
     26453952x_2^8x_5-2420208x_2^8+11337408x_2^7x_5^2+6223392x_
     ^2x_5+9110047128731136x_1x_2x_5^5-1250184222455616x_1x_2x_
                                                               
     ------------------------------------------------------------------------
                                                           
     1285956x_2^5x_5^4+1653372x_2^4x_5^5+2250423x_2^2x_5^6+
     2^7x_5-12002256x_2^6x_5^2+15431472x_2^5x_5^3-19840464x
     5^4+686257960245984x_1x_2x_5^3+282528354346812x_1x_2x_
                                                           
     ------------------------------------------------------------------------
                                                   
     2893401x_2x_5^7                               
     _2^4x_5^4+5445468x_2^4x_5^3+5764801x_2^3x_5^3-
     5^2-18894912563294208x_2^9+12146729504974848x_
                                                   
     ------------------------------------------------------------------------
                                                            
                                                            
     27005076x_2^2x_5^5+14823774x_2^2x_5^4-34720812x_2x_5^6+
     2^8x_5+1666912296607488x_2^8-5205741216417792x_2^7x_5^2
                                                            
     ------------------------------------------------------------------------
                                                                             
                                                                             
     9529569x_2x_5^5                                                         
     -3571954921301760x_2^7x_5+196073702927424x_2^7+5511016164294144x_2^6x_5^
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     2-756284282720064x_2^6x_5-207571852173168x_2^6-7085592211235328x_2^5x_5^
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     3+972365506354368x_2^5x_5^2+266878095651216x_2^5x_5+219744275603076x_2^5
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     +9110047128731136x_2^4x_5^4-1250184222455616x_2^4x_5^3+686257960245984x_
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     2^4x_5^2+282528354346812x_2^4x_5+232630513987207x_2^4+1089752223909132x_
                                                                             
     ------------------------------------------------------------------------
                                                                  
                                                                  
                                                                  
     2^3x_5^2+897289125379227x_2^3x_5+12399786369661824x_2^2x_5^5-
                                                                  
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     1701639636120144x_2^2x_5^4+2335183336948140x_2^2x_5^3+1153657446916149x_
                                                                             
     ------------------------------------------------------------------------
                                                                 
                                                                 
                                                                 
     2^2x_5^2+15942582475279488x_2x_5^6-2187822389297328x_2x_5^5+
                                                                 
     ------------------------------------------------------------------------
                                                      |
                                                      |
                                                      |
     1200951430430472x_2x_5^4+494424620106921x_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

                      1             1     5                        2   1    
o13 = (map(R,R,{4x  + -x  + x , x , -x  + -x  + x , x }), ideal (5x  + -x x 
                  1   2 2    4   1  6 1   4 2    3   2             1   2 1 2
      -----------------------------------------------------------------------
                  2 3     61 2 2   5   3     2       1   2     1 2      
      + x x  + 1, -x x  + --x x  + -x x  + 4x x x  + -x x x  + -x x x  +
         1 4      3 1 2   12 1 2   8 1 2     1 2 3   2 1 2 3   6 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

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

                9     3                                          16 2   3    
o16 = (map(R,R,{-x  + -x  + x , x , 8x  + 3x  + x , x }), ideal (--x  + -x x 
                7 1   2 2    4   1    1     2    3   2            7 1   2 1 2
      -----------------------------------------------------------------------
                  72 3     111 2 2   9   3   9 2       3   2       2      
      + x x  + 1, --x x  + ---x x  + -x x  + -x x x  + -x x x  + 8x x x  +
         1 4       7 1 2    7  1 2   2 1 2   7 1 2 3   2 1 2 3     1 2 4  
      -----------------------------------------------------------------------
          2
      3x x x  + x x x x  + 1), {x , x })
        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

                                                                    2        
o19 = (map(R,R,{2x  - 2x  + x , x , - 2x  + x  + x , x }), ideal (3x  - 2x x 
                  1     2    4   1      1    2    3   2             1     1 2
      -----------------------------------------------------------------------
                      3       2 2       3     2           2       2      
      + x x  + 1, - 4x x  + 6x x  - 2x x  + 2x x x  - 2x x x  - 2x x x  +
         1 4          1 2     1 2     1 2     1 2 3     1 2 3     1 2 4  
      -----------------------------------------------------------------------
         2
      x x x  + x x x x  + 1), {x , x })
       1 2 4    1 2 3 4         4   3

o19 : Sequence

This symbol is provided by the package NoetherNormalization.

Ways to use noetherNormalization :