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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

                                        6                        2          
o3 = (map(R,R,{x  + 4x  + x , x , 2x  + -x  + x , x }), ideal (2x  + 4x x  +
                1     2    4   1    1   7 2    3   2             1     1 2  
     ------------------------------------------------------------------------
                 3     62 2 2   24   3    2           2       2       6   2
     x x  + 1, 2x x  + --x x  + --x x  + x x x  + 4x x x  + 2x x x  + -x x x 
      1 4        1 2    7 1 2    7 1 2    1 2 3     1 2 3     1 2 4   7 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     9                   5         1     1              
o6 = (map(R,R,{-x  + -x  + x , x , 4x  + -x  + x , -x  + -x  + x , x }),
               9 1   2 2    5   1    1   9 2    4  9 1   9 2    3   2   
     ------------------------------------------------------------------------
            4 2   9               3   64 3     8 2 2   16 2            3  
     ideal (-x  + -x x  + x x  - x , ---x x  + -x x  + --x x x  + 27x x  +
            9 1   2 1 2    1 5    2  729 1 2   3 1 2   27 1 2 5      1 2  
     ------------------------------------------------------------------------
          2     4     2   729 4   243 3     27 2 2      3
     12x x x  + -x x x  + ---x  + ---x x  + --x x  + x x ), {x , x , x })
        1 2 5   3 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} | 288x_1x_2x_5^6-15552x_2^9x_5-531441x_2^9+1728x_2
     {-9}  | 1417176x_1x_2^2x_5^3-4608x_1x_2x_5^5+314928x_1x_
     {-9}  | 183014339639688x_1x_2^3+595077871104x_1x_2^2x_5^
     {-3}  | 8x_1^2+81x_1x_2+18x_1x_5-18x_2^3                
     ------------------------------------------------------------------------
                                                                         
     ^8x_5^2+118098x_2^8x_5-128x_2^7x_5^3-26244x_2^7x_5^2+5832x_2^6x_5^3-
     2x_5^4+248832x_2^9-27648x_2^8x_5-629856x_2^8+2048x_2^7x_5^2+279936x_
     2+81339706506528x_1x_2^2x_5+18874368x_1x_2x_5^5-644972544x_1x_2x_5^4
                                                                         
     ------------------------------------------------------------------------
                                                                     
     1296x_2^5x_5^4+288x_2^4x_5^5+2916x_2^2x_5^6+648x_2x_5^7         
     2^7x_5-93312x_2^6x_5^2+20736x_2^5x_5^3-4608x_2^4x_5^4+314928x_2^
     +88159684608x_1x_2x_5^3+9037745167392x_1x_2x_5^2-1019215872x_2^9
                                                                     
     ------------------------------------------------------------------------
                                                                             
                                                                             
     4x_5^3+14348907x_2^3x_5^3-46656x_2^2x_5^5+6377292x_2^2x_5^4-10368x_2x_5^
     +113246208x_2^8x_5+3869835264x_2^8-8388608x_2^7x_5^2-1433272320x_2^7x_5+
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
     6+708588x_2x_5^5                                                        
     19591041024x_2^7+382205952x_2^6x_5^2-13060694016x_2^6x_5-892616806656x_2
                                                                             
     ------------------------------------------------------------------------
                                                                     
                                                                     
                                                                     
     ^6-84934656x_2^5x_5^3+2902376448x_2^5x_5^2+198359290368x_2^5x_5+
                                                                     
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     40669853253264x_2^5+18874368x_2^4x_5^4-644972544x_2^4x_5^3+88159684608x_
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     2^4x_5^2+9037745167392x_2^4x_5+1853020188851841x_2^4+6025163444928x_2^3x
                                                                             
     ------------------------------------------------------------------------
                                                                            
                                                                            
                                                                            
     _5^2+1235346792567894x_2^3x_5+191102976x_2^2x_5^5-6530347008x_2^2x_5^4+
                                                                            
     ------------------------------------------------------------------------
                                                                        
                                                                        
                                                                        
     2231542016640x_2^2x_5^3+274521509459532x_2^2x_5^2+42467328x_2x_5^6-
                                                                        
     ------------------------------------------------------------------------
                                                                    |
                                                                    |
                                                                    |
     1451188224x_2x_5^5+198359290368x_2x_5^4+20334926626632x_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

                2                    3                            7 2  
o13 = (map(R,R,{-x  + 10x  + x , x , -x  + 4x  + x , x }), ideal (-x  +
                5 1      2    4   1  5 1     2    3   2           5 1  
      -----------------------------------------------------------------------
                          6 3     38 2 2        3   2 2            2    
      10x x  + x x  + 1, --x x  + --x x  + 40x x  + -x x x  + 10x x x  +
         1 2    1 4      25 1 2    5 1 2      1 2   5 1 2 3      1 2 3  
      -----------------------------------------------------------------------
      3 2           2
      -x x x  + 4x x x  + x x x x  + 1), {x , x })
      5 1 2 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

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

o19 : Sequence

This symbol is provided by the package NoetherNormalization.

Ways to use noetherNormalization :