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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

               5     2             7                            11 2   2    
o3 = (map(R,R,{-x  + -x  + x , x , -x  + 2x  + x , x }), ideal (--x  + -x x 
               6 1   5 2    4   1  4 1     2    3   2            6 1   5 1 2
     ------------------------------------------------------------------------
                 35 3     71 2 2   4   3   5 2       2   2     7 2      
     + x x  + 1, --x x  + --x x  + -x x  + -x x x  + -x x x  + -x x x  +
        1 4      24 1 2   30 1 2   5 1 2   6 1 2 3   5 1 2 3   4 1 2 4  
     ------------------------------------------------------------------------
         2
     2x x x  + x x x x  + 1), {x , x })
       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     5             1               5     1              
o6 = (map(R,R,{-x  + -x  + x , x , -x  + 3x  + x , -x  + -x  + x , x }),
               5 1   4 2    5   1  4 1     2    4  4 1   9 2    3   2   
     ------------------------------------------------------------------------
            9 2   5               3  729 3     243 2 2   243 2       135   3
     ideal (-x  + -x x  + x x  - x , ---x x  + ---x x  + ---x x x  + ---x x 
            5 1   4 1 2    1 5    2  125 1 2    20 1 2    25 1 2 5    16 1 2
     ------------------------------------------------------------------------
       27   2     27     2   125 4   75 3     15 2 2      3
     + --x x x  + --x x x  + ---x  + --x x  + --x x  + x x ), {x , x , x })
        2 1 2 5    5 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} | 46080x_1x_2x_5^6-777600x_2^9x_5-140625x
     {-9}  | 562500x_1x_2^2x_5^3-1244160x_1x_2x_5^5+
     {-9}  | 1098632812500x_1x_2^3+2430000000000x_1x
     {-3}  | 36x_1^2+25x_1x_2+20x_1x_5-20x_2^3      
     ------------------------------------------------------------------------
                                                                 
     _2^9+311040x_2^8x_5^2+112500x_2^8x_5-82944x_2^7x_5^3-90000x_
     450000x_1x_2x_5^4+20995200x_2^9-8398080x_2^8x_5-1012500x_2^8
     _2^2x_5^2+1757812500000x_1x_2^2x_5+9906778275840x_1x_2x_5^5-
                                                                 
     ------------------------------------------------------------------------
                                                                            
     2^7x_5^2+72000x_2^6x_5^3-57600x_2^5x_5^4+46080x_2^4x_5^5+32000x_2^2x_5^
     +2239488x_2^7x_5^2+1620000x_2^7x_5-1944000x_2^6x_5^2+1555200x_2^5x_5^3-
     1791590400000x_1x_2x_5^4+1296000000000x_1x_2x_5^3+703125000000x_1x_2x_5
                                                                            
     ------------------------------------------------------------------------
                                                                        
     6+25600x_2x_5^7                                                    
     1244160x_2^4x_5^4+450000x_2^4x_5^3+390625x_2^3x_5^3-864000x_2^2x_5^
     ^2-167176883404800x_2^9+66870753361920x_2^8x_5+12093235200000x_2^8-
                                                                        
     ------------------------------------------------------------------------
                                                                        
                                                                        
     5+625000x_2^2x_5^4-691200x_2x_5^6+250000x_2x_5^5                   
     17832200896512x_2^7x_5^2-16124313600000x_2^7x_5+1166400000000x_2^7+
                                                                        
     ------------------------------------------------------------------------
                                                                       
                                                                       
                                                                       
     15479341056000x_2^6x_5^2-2799360000000x_2^6x_5-1012500000000x_2^6-
                                                                       
     ------------------------------------------------------------------------
                                                                           
                                                                           
                                                                           
     12383472844800x_2^5x_5^3+2239488000000x_2^5x_5^2+810000000000x_2^5x_5+
                                                                           
     ------------------------------------------------------------------------
                                                                       
                                                                       
                                                                       
     878906250000x_2^5+9906778275840x_2^4x_5^4-1791590400000x_2^4x_5^3+
                                                                       
     ------------------------------------------------------------------------
                                                                    
                                                                    
                                                                    
     1296000000000x_2^4x_5^2+703125000000x_2^4x_5+762939453125x_2^4+
                                                                    
     ------------------------------------------------------------------------
                                                                           
                                                                           
                                                                           
     1687500000000x_2^3x_5^2+1831054687500x_2^3x_5+6879707136000x_2^2x_5^5-
                                                                           
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     1244160000000x_2^2x_5^4+2250000000000x_2^2x_5^3+1464843750000x_2^2x_5^2+
                                                                             
     ------------------------------------------------------------------------
                                                                     
                                                                     
                                                                     
     5503765708800x_2x_5^6-995328000000x_2x_5^5+720000000000x_2x_5^4+
                                                                     
     ------------------------------------------------------------------------
                          |
                          |
                          |
     390625000000x_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

                     2             6     5                        2   2      
o13 = (map(R,R,{x  + -x  + x , x , -x  + -x  + x , x }), ideal (2x  + -x x  +
                 1   9 2    4   1  7 1   6 2    3   2             1   9 1 2  
      -----------------------------------------------------------------------
                6 3     43 2 2    5   3    2       2   2     6 2      
      x x  + 1, -x x  + --x x  + --x x  + x x x  + -x x x  + -x x x  +
       1 4      7 1 2   42 1 2   27 1 2    1 2 3   9 1 2 3   7 1 2 4  
      -----------------------------------------------------------------------
      5   2
      -x x x  + x x x x  + 1), {x , x })
      6 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

                1                   7                           3 2          
o16 = (map(R,R,{-x  + 2x  + x , x , -x  + x  + x , x }), ideal (-x  + 2x x  +
                2 1     2    4   1  3 1    2    3   2           2 1     1 2  
      -----------------------------------------------------------------------
                7 3     31 2 2       3   1 2           2     7 2          2
      x x  + 1, -x x  + --x x  + 2x x  + -x x x  + 2x x x  + -x x x  + x x x 
       1 4      6 1 2    6 1 2     1 2   2 1 2 3     1 2 3   3 1 2 4    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
--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
--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: 3
--trying with basis element limit: 5
--trying with basis element limit: 20

                                                                       2  
o19 = (map(R,R,{- 5x  + 2x  + x , x , 2x  - 2x  + x , x }), ideal (- 4x  +
                    1     2    4   1    1     2    3   2               1  
      -----------------------------------------------------------------------
                             3        2 2       3     2           2    
      2x x  + x x  + 1, - 10x x  + 14x x  - 4x x  - 5x x x  + 2x x x  +
        1 2    1 4           1 2      1 2     1 2     1 2 3     1 2 3  
      -----------------------------------------------------------------------
        2           2
      2x 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 :