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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

               4                                               7 2         
o3 = (map(R,R,{-x  + x  + x , x , 2x  + 2x  + x , x }), ideal (-x  + x x  +
               3 1    2    4   1    1     2    3   2           3 1    1 2  
     ------------------------------------------------------------------------
               8 3     14 2 2       3   4 2          2       2           2
     x x  + 1, -x x  + --x x  + 2x x  + -x x x  + x x x  + 2x x x  + 2x x x 
      1 4      3 1 2    3 1 2     1 2   3 1 2 3    1 2 3     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

               7     3                  3               7                    
o6 = (map(R,R,{-x  + -x  + x , x , x  + -x  + x , 3x  + -x  + x , x }), ideal
               6 1   4 2    5   1   1   7 2    4    1   8 2    3   2         
     ------------------------------------------------------------------------
      7 2   3               3  343 3     49 2 2   49 2       63   3  
     (-x  + -x x  + x x  - x , ---x x  + --x x  + --x x x  + --x x  +
      6 1   4 1 2    1 5    2  216 1 2   16 1 2   12 1 2 5   32 1 2  
     ------------------------------------------------------------------------
     21   2     7     2   27 4   27 3     9 2 2      3
     --x x x  + -x x x  + --x  + --x x  + -x x  + x x ), {x , x , x })
      4 1 2 5   2 1 2 5   64 2   16 2 5   4 2 5    2 5     5   4   3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10                                            
     {-10} | 21504x_1x_2x_5^6-84672x_2^9x_5-5103x_2^9+56448x_2^
     {-9}  | 3402x_1x_2^2x_5^3-37632x_1x_2x_5^5+4536x_1x_2x_5^4
     {-9}  | 22320522x_1x_2^3+246903552x_1x_2^2x_5^2+59521392x_
     {-3}  | 14x_1^2+9x_1x_2+12x_1x_5-12x_2^3                  
     ------------------------------------------------------------------------
                                                                        
     8x_5^2+6804x_2^8x_5-25088x_2^7x_5^3-9072x_2^7x_5^2+12096x_2^6x_5^3-
     +148176x_2^9-98784x_2^8x_5-3969x_2^8+43904x_2^7x_5^2+10584x_2^7x_5-
     1x_2^2x_5+15105785856x_1x_2x_5^5-910393344x_1x_2x_5^4+219469824x_1x
                                                                        
     ------------------------------------------------------------------------
                                                                    
     16128x_2^5x_5^4+21504x_2^4x_5^5+13824x_2^2x_5^6+18432x_2x_5^7  
     21168x_2^6x_5^2+28224x_2^5x_5^3-37632x_2^4x_5^4+4536x_2^4x_5^3+
     _2x_5^3+39680928x_1x_2x_5^2-59479031808x_2^9+39652687872x_2^8x_
                                                                    
     ------------------------------------------------------------------------
                                                                             
                                                                             
     2187x_2^3x_5^3-24192x_2^2x_5^5+5832x_2^2x_5^4-32256x_2x_5^6+3888x_2x_5^5
     5+2389782528x_2^8-17623416832x_2^7x_5^2-5310627840x_2^7x_5+128024064x_2^
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     7+8497004544x_2^6x_5^2-512096256x_2^6x_5-61725888x_2^6-11329339392x_2^5x
                                                                             
     ------------------------------------------------------------------------
                                                                            
                                                                            
                                                                            
     _5^3+682795008x_2^5x_5^2+82301184x_2^5x_5+29760696x_2^5+15105785856x_2^
                                                                            
     ------------------------------------------------------------------------
                                                                     
                                                                     
                                                                     
     4x_5^4-910393344x_2^4x_5^3+219469824x_2^4x_5^2+39680928x_2^4x_5+
                                                                     
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     14348907x_2^4+158723712x_2^3x_5^2+57395628x_2^3x_5+9710862336x_2^2x_5^5-
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     585252864x_2^2x_5^4+352719360x_2^2x_5^3+76527504x_2^2x_5^2+12947816448x_
                                                                             
     ------------------------------------------------------------------------
                                                                 |
                                                                 |
                                                                 |
     2x_5^6-780337152x_2x_5^5+188116992x_2x_5^4+34012224x_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

                      7             2     7                        2   7    
o13 = (map(R,R,{3x  + -x  + x , x , -x  + -x  + x , x }), ideal (4x  + -x x 
                  1   9 2    4   1  9 1   5 2    3   2             1   9 1 2
      -----------------------------------------------------------------------
                  2 3     1771 2 2   49   3     2       7   2     2 2      
      + x x  + 1, -x x  + ----x x  + --x x  + 3x x x  + -x x x  + -x x x  +
         1 4      3 1 2    405 1 2   45 1 2     1 2 3   9 1 2 3   9 1 2 4  
      -----------------------------------------------------------------------
      7   2
      -x x x  + x x x x  + 1), {x , x })
      5 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

                 3     2             7     5                      13 2  
o16 = (map(R,R,{--x  + -x  + x , x , -x  + -x  + x , x }), ideal (--x  +
                10 1   7 2    4   1  6 1   9 2    3   2           10 1  
      -----------------------------------------------------------------------
      2                  7 3     1 2 2   10   3    3 2       2   2    
      -x x  + x x  + 1, --x x  + -x x  + --x x  + --x x x  + -x x x  +
      7 1 2    1 4      20 1 2   2 1 2   63 1 2   10 1 2 3   7 1 2 3  
      -----------------------------------------------------------------------
      7 2       5   2
      -x x x  + -x x x  + x x x x  + 1), {x , x })
      6 1 2 4   9 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,{x  + 2x  + x , x , 2x  - x  + x , x }), ideal (2x  + 2x x  +
                 1     2    4   1    1    2    3   2             1     1 2  
      -----------------------------------------------------------------------
                  3       2 2       3    2           2       2          2
      x x  + 1, 2x x  + 3x x  - 2x x  + x x x  + 2x x x  + 2x x x  - x x x  +
       1 4        1 2     1 2     1 2    1 2 3     1 2 3     1 2 4    1 2 4  
      -----------------------------------------------------------------------
      x x x x  + 1), {x , x })
       1 2 3 4         4   3

o19 : Sequence

This symbol is provided by the package NoetherNormalization.

Ways to use noetherNormalization :