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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

               4     3             7                            9 2   3      
o3 = (map(R,R,{-x  + -x  + x , x , -x  + 7x  + x , x }), ideal (-x  + -x x  +
               5 1   7 2    4   1  2 1     2    3   2           5 1   7 1 2  
     ------------------------------------------------------------------------
               14 3     71 2 2       3   4 2       3   2     7 2      
     x x  + 1, --x x  + --x x  + 3x x  + -x x x  + -x x x  + -x x x  +
      1 4       5 1 2   10 1 2     1 2   5 1 2 3   7 1 2 3   2 1 2 4  
     ------------------------------------------------------------------------
         2
     7x 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

               10     7             9     3               5              
o6 = (map(R,R,{--x  + -x  + x , x , -x  + -x  + x , 5x  + -x  + x , x }),
                9 1   8 2    5   1  2 1   4 2    4    1   3 2    3   2   
     ------------------------------------------------------------------------
            10 2   7               3  1000 3     175 2 2   100 2      
     ideal (--x  + -x x  + x x  - x , ----x x  + ---x x  + ---x x x  +
             9 1   8 1 2    1 5    2   729 1 2    54 1 2    27 1 2 5  
     ------------------------------------------------------------------------
     245   3   35   2     10     2   343 4   147 3     21 2 2      3
     ---x x  + --x x x  + --x x x  + ---x  + ---x x  + --x x  + x x ), {x ,
      96 1 2    6 1 2 5    3 1 2 5   512 2    64 2 5    8 2 5    2 5     5 
     ------------------------------------------------------------------------
     x , x })
      4   3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10           
     {-10} | 1474560x_1x_2x_5^
     {-9}  | 5186160x_1x_2^2x_
     {-9}  | 115317695651760x_
     {-3}  | 80x_1^2+63x_1x_2+
     ------------------------------------------------------------------------
                                                               
     6-7526400x_2^9x_5-756315x_2^9+4300800x_2^8x_5^2+864360x_2^
     5^3-29491200x_1x_2x_5^5+5927040x_1x_2x_5^4+150528000x_2^9-
     1x_2^3+655756325683200x_1x_2^2x_5^2+263583304346880x_1x_2^
     72x_1x_5-72x_2^3                                          
     ------------------------------------------------------------------------
                                                               
     8x_5-1638400x_2^7x_5^3-987840x_2^7x_5^2+1128960x_2^6x_5^3-
     86016000x_2^8x_5-5762400x_2^8+32768000x_2^7x_5^2+13171200x
     2x_5+12369505812480000x_1x_2x_5^5-1242990379008000x_1x_2x_
                                                               
     ------------------------------------------------------------------------
                                
     1290240x_2^5x_5^4+1474560x_
     _2^7x_5-22579200x_2^6x_5^2+
     5^4+499623867187200x_1x_2x_
                                
     ------------------------------------------------------------------------
                                                                             
     2^4x_5^5+1161216x_2^2x_5^6+1327104x_2x_5^7                              
     25804800x_2^5x_5^3-29491200x_2^4x_5^4+5927040x_2^4x_5^3+4084101x_2^3x_5^
     5^3+150619031055360x_1x_2x_5^2-63136019251200000x_2^9+36077725286400000x
                                                                             
     ------------------------------------------------------------------------
                                                             
                                                             
     3-23224320x_2^2x_5^5+9335088x_2^2x_5^4-26542080x_2x_5^6+
     _2^8x_5+3625388605440000x_2^8-13743895347200000x_2^7x_5^
                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
     5334336x_2x_5^5                                                         
     2-6905502105600000x_2^7x_5+277568815104000x_2^7+9470402887680000x_2^6x_5
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     ^2-951664508928000x_2^6x_5-191262261657600x_2^6-10823317585920000x_2^5x_
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     5^3+1087616581632000x_2^5x_5^2+218585441894400x_2^5x_5+131791652173440x_
                                                                             
     ------------------------------------------------------------------------
                                                                
                                                                
                                                                
     2^5+12369505812480000x_2^4x_5^4-1242990379008000x_2^4x_5^3+
                                                                
     ------------------------------------------------------------------------
                                                                           
                                                                           
                                                                           
     499623867187200x_2^4x_5^2+150619031055360x_2^4x_5+90812685325761x_2^4+
                                                                           
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     516408106475520x_2^3x_5^2+311357778259752x_2^3x_5+9740985827328000x_2^2x
                                                                             
     ------------------------------------------------------------------------
                                                              
                                                              
                                                              
     _5^5-978854923468800x_2^2x_5^4+983634488524800x_2^2x_5^3+
                                                              
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     355837460868288x_2^2x_5^2+11132555231232000x_2x_5^6-1118691341107200x_2x
                                                                             
     ------------------------------------------------------------------------
                                                          |
                                                          |
                                                          |
     _5^5+449661480468480x_2x_5^4+135557127949824x_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

                                    3      7                        2        
o13 = (map(R,R,{x  + 4x  + x , x , --x  + --x  + x , x }), ideal (2x  + 4x x 
                 1     2    4   1  10 1   10 2    3   2             1     1 2
      -----------------------------------------------------------------------
                   3 3     19 2 2   14   3    2           2      3 2      
      + x x  + 1, --x x  + --x x  + --x x  + x x x  + 4x x x  + --x x x  +
         1 4      10 1 2   10 1 2    5 1 2    1 2 3     1 2 3   10 1 2 4  
      -----------------------------------------------------------------------
       7   2
      --x x x  + x x x x  + 1), {x , x })
      10 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     8             4     3                      6 2   8    
o16 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (-x  + -x x 
                5 1   5 2    4   1  5 1   7 2    3   2           5 1   5 1 2
      -----------------------------------------------------------------------
                   4 3     239 2 2   24   3   1 2       8   2     4 2      
      + x x  + 1, --x x  + ---x x  + --x x  + -x x x  + -x x x  + -x x x  +
         1 4      25 1 2   175 1 2   35 1 2   5 1 2 3   5 1 2 3   5 1 2 4  
      -----------------------------------------------------------------------
      3   2
      -x x x  + x x x x  + 1), {x , x })
      7 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
--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

                                                                     2  
o19 = (map(R,R,{4x  + 2x  + x , x , - 4x  - 2x  + x , x }), ideal (5x  +
                  1     2    4   1      1     2    3   2             1  
      -----------------------------------------------------------------------
                             3        2 2       3     2           2    
      2x x  + x x  + 1, - 16x x  - 16x x  - 4x x  + 4x x x  + 2x x x  -
        1 2    1 4           1 2      1 2     1 2     1 2 3     1 2 3  
      -----------------------------------------------------------------------
        2           2
      4x 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 :