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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

               2     5             7     9                      7 2   5      
o3 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (-x  + -x x  +
               5 1   4 2    4   1  6 1   2 2    3   2           5 1   4 1 2  
     ------------------------------------------------------------------------
                7 3     391 2 2   45   3   2 2       5   2     7 2      
     x x  + 1, --x x  + ---x x  + --x x  + -x x x  + -x x x  + -x x x  +
      1 4      15 1 2   120 1 2    8 1 2   5 1 2 3   4 1 2 3   6 1 2 4  
     ------------------------------------------------------------------------
     9   2
     -x x x  + x x x x  + 1), {x , x })
     2 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

               1     7             10     6         7     3              
o6 = (map(R,R,{-x  + -x  + x , x , --x  + -x  + x , -x  + -x  + x , x }),
               4 1   6 2    5   1   3 1   5 2    4  4 1   4 2    3   2   
     ------------------------------------------------------------------------
            1 2   7               3   1 3      7 2 2    3 2       49   3  
     ideal (-x  + -x x  + x x  - x , --x x  + --x x  + --x x x  + --x x  +
            4 1   6 1 2    1 5    2  64 1 2   32 1 2   16 1 2 5   48 1 2  
     ------------------------------------------------------------------------
     7   2     3     2   343 4   49 3     7 2 2      3
     -x x x  + -x x x  + ---x  + --x x  + -x x  + x x ), {x , x , x })
     4 1 2 5   4 1 2 5   216 2   12 2 5   2 2 5    2 5     5   4   3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10                                                          
     {-10} | 7776x_1x_2x_5^6-15876x_2^9x_5-16807x_2^9+6804x_2^8x_5^2+14406x_2
     {-9}  | 28812x_1x_2^2x_5^3-11664x_1x_2x_5^5+24696x_1x_2x_5^4+23814x_2^9-
     {-9}  | 5931980229x_1x_2^3+2401451388x_1x_2^2x_5^2+10169108964x_1x_2^2x_
     {-3}  | 3x_1^2+14x_1x_2+12x_1x_5-12x_2^3                                
     ------------------------------------------------------------------------
                                                                       
     ^8x_5-1944x_2^7x_5^3-12348x_2^7x_5^2+10584x_2^6x_5^3-9072x_2^5x_5^
     10206x_2^8x_5-7203x_2^8+2916x_2^7x_5^2+12348x_2^7x_5-15876x_2^6x_5
     5+306110016x_1x_2x_5^5-324060912x_1x_2x_5^4+1372257936x_1x_2x_5^3+
                                                                       
     ------------------------------------------------------------------------
                                                                         
     4+7776x_2^4x_5^5+36288x_2^2x_5^6+31104x_2x_5^7                      
     ^2+13608x_2^5x_5^3-11664x_2^4x_5^4+24696x_2^4x_5^3+134456x_2^3x_5^3-
     4358189556x_1x_2x_5^2-624974616x_2^9+267846264x_2^8x_5+283553298x_2^
                                                                         
     ------------------------------------------------------------------------
                                                                             
                                                                             
     54432x_2^2x_5^5+230496x_2^2x_5^4-46656x_2x_5^6+98784x_2x_5^5            
     8-76527504x_2^7x_5^2-405076140x_2^7x_5+171532242x_2^7+416649744x_2^6x_5^
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     2-441082908x_2^6x_5-933897762x_2^6-357128352x_2^5x_5^3+378071064x_2^5x_5
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     ^2+800483796x_2^5x_5+5084554482x_2^5+306110016x_2^4x_5^4-324060912x_2^4x
                                                                             
     ------------------------------------------------------------------------
                                                                   
                                                                   
                                                                   
     _5^3+1372257936x_2^4x_5^2+4358189556x_2^4x_5+27682574402x_2^4+
                                                                   
     ------------------------------------------------------------------------
                                                                    
                                                                    
                                                                    
     11206773144x_2^3x_5^2+71183762748x_2^3x_5+1428513408x_2^2x_5^5-
                                                                    
     ------------------------------------------------------------------------
                                                                      
                                                                      
                                                                      
     1512284256x_2^2x_5^4+16009675920x_2^2x_5^3+61014653784x_2^2x_5^2+
                                                                      
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     1224440064x_2x_5^6-1296243648x_2x_5^5+5489031744x_2x_5^4+17432758224x_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,{a + 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

                9                   3     6                      17 2        
o13 = (map(R,R,{-x  + 8x  + x , x , -x  + -x  + x , x }), ideal (--x  + 8x x 
                8 1     2    4   1  4 1   7 2    3   2            8 1     1 2
      -----------------------------------------------------------------------
                  27 3     195 2 2   48   3   9 2           2     3 2      
      + x x  + 1, --x x  + ---x x  + --x x  + -x x x  + 8x x x  + -x x x  +
         1 4      32 1 2    28 1 2    7 1 2   8 1 2 3     1 2 3   4 1 2 4  
      -----------------------------------------------------------------------
      6   2
      -x x x  + x x x x  + 1), {x , x })
      7 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

                4     8              9     1                      7 2   8    
o16 = (map(R,R,{-x  + -x  + x , x , --x  + -x  + x , x }), ideal (-x  + -x x 
                3 1   7 2    4   1  10 1   2 2    3   2           3 1   7 1 2
      -----------------------------------------------------------------------
                  6 3     178 2 2   4   3   4 2       8   2      9 2      
      + x x  + 1, -x x  + ---x x  + -x x  + -x x x  + -x x x  + --x x x  +
         1 4      5 1 2   105 1 2   7 1 2   3 1 2 3   7 1 2 3   10 1 2 4  
      -----------------------------------------------------------------------
      1   2
      -x x x  + x x x x  + 1), {x , x })
      2 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 :