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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

               3     9             7     2                      5 2   9      
o3 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (-x  + -x x  +
               2 1   7 2    4   1  6 1   5 2    3   2           2 1   7 1 2  
     ------------------------------------------------------------------------
               7 3     21 2 2   18   3   3 2       9   2     7 2      
     x x  + 1, -x x  + --x x  + --x x  + -x x x  + -x x x  + -x x x  +
      1 4      4 1 2   10 1 2   35 1 2   2 1 2 3   7 1 2 3   6 1 2 4  
     ------------------------------------------------------------------------
     2   2
     -x x x  + x x x x  + 1), {x , x })
     5 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

               2     6             5              4     5                    
o6 = (map(R,R,{-x  + -x  + x , x , -x  + x  + x , -x  + -x  + x , x }), ideal
               9 1   7 2    5   1  7 1    2    4  9 1   2 2    3   2         
     ------------------------------------------------------------------------
      2 2   6               3   8  3      8 2 2    4 2       24   3   8   2  
     (-x  + -x x  + x x  - x , ---x x  + --x x  + --x x x  + --x x  + -x x x 
      9 1   7 1 2    1 5    2  729 1 2   63 1 2   27 1 2 5   49 1 2   7 1 2 5
     ------------------------------------------------------------------------
       2     2   216 4   108 3     18 2 2      3
     + -x x x  + ---x  + ---x x  + --x x  + x x ), {x , x , x })
       3 1 2 5   343 2    49 2 5    7 2 5    2 5     5   4   3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10                                       
     {-10} | 302526x_1x_2x_5^6-296352x_2^9x_5-139968x_2^9+
     {-9}  | 244944x_1x_2^2x_5^3-302526x_1x_2x_5^5+285768x
     {-9}  | 231419172096x_1x_2^3+285821724384x_1x_2^2x_5^
     {-3}  | 14x_1^2+54x_1x_2+63x_1x_5-63x_2^3            
     ------------------------------------------------------------------------
                                                                             
     172872x_2^8x_5^2+163296x_2^8x_5-67228x_2^7x_5^3-190512x_2^7x_5^2+222264x
     _1x_2x_5^4+296352x_2^9-172872x_2^8x_5-54432x_2^8+67228x_2^7x_5^2+127008x
     2+539978068224x_1x_2^2x_5+249143169618x_1x_2x_5^5-117671118012x_1x_2x_5^
                                                                             
     ------------------------------------------------------------------------
                                               
     _2^6x_5^3-259308x_2^5x_5^4+302526x_2^4x_5^
     _2^7x_5-222264x_2^6x_5^2+259308x_2^5x_5^3-
     4+222305785632x_1x_2x_5^3+314987206464x_1x
                                               
     ------------------------------------------------------------------------
                                                                     
     5+1166886x_2^2x_5^6+1361367x_2x_5^7                             
     302526x_2^4x_5^4+285768x_2^4x_5^3+944784x_2^3x_5^3-1166886x_2^2x
     _2x_5^2-244058615136x_2^9+142367525496x_2^8x_5+67240638864x_2^8-
                                                                     
     ------------------------------------------------------------------------
                                                                 
                                                                 
     _5^5+2204496x_2^2x_5^4-1361367x_2x_5^6+1285956x_2x_5^5      
     55365148804x_2^7x_5^2-130745686680x_2^7x_5+24700642848x_2^7+
                                                                 
     ------------------------------------------------------------------------
                                                                 
                                                                 
                                                                 
     183043961352x_2^6x_5^2-86452249968x_2^6x_5-81663349824x_2^6-
                                                                 
     ------------------------------------------------------------------------
                                                                       
                                                                       
                                                                       
     213551288244x_2^5x_5^3+100860958296x_2^5x_5^2+95273908128x_2^5x_5+
                                                                       
     ------------------------------------------------------------------------
                                                                     
                                                                     
                                                                     
     269989034112x_2^5+249143169618x_2^4x_5^4-117671118012x_2^4x_5^3+
                                                                     
     ------------------------------------------------------------------------
                                                                   
                                                                   
                                                                   
     222305785632x_2^4x_5^2+314987206464x_2^4x_5+892616806656x_2^4+
                                                                   
     ------------------------------------------------------------------------
                                                                          
                                                                          
                                                                          
     1102455222624x_2^3x_5^2+3124158823296x_2^3x_5+960980797098x_2^2x_5^5-
                                                                          
     ------------------------------------------------------------------------
                                                                            
                                                                            
                                                                            
     453874312332x_2^2x_5^4+2143662932880x_2^2x_5^3+3644851960512x_2^2x_5^2+
                                                                            
     ------------------------------------------------------------------------
                                                                      
                                                                      
                                                                      
     1121144263281x_2x_5^6-529520031054x_2x_5^5+1000376035344x_2x_5^4+
                                                                      
     ------------------------------------------------------------------------
                           |
                           |
                           |
     1417442429088x_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

                10                        1                      13 2        
o13 = (map(R,R,{--x  + 7x  + x , x , x  + -x  + x , x }), ideal (--x  + 7x x 
                 3 1     2    4   1   1   2 2    3   2            3 1     1 2
      -----------------------------------------------------------------------
                  10 3     26 2 2   7   3   10 2           2      2      
      + x x  + 1, --x x  + --x x  + -x x  + --x x x  + 7x x x  + x x x  +
         1 4       3 1 2    3 1 2   2 1 2    3 1 2 3     1 2 3    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

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

                8                         5                      11 2        
o16 = (map(R,R,{-x  + 5x  + x , x , 5x  + -x  + x , x }), ideal (--x  + 5x x 
                3 1     2    4   1    1   8 2    3   2            3 1     1 2
      -----------------------------------------------------------------------
                  40 3     80 2 2   25   3   8 2           2       2      
      + x x  + 1, --x x  + --x x  + --x x  + -x x x  + 5x x x  + 5x x x  +
         1 4       3 1 2    3 1 2    8 1 2   3 1 2 3     1 2 3     1 2 4  
      -----------------------------------------------------------------------
      5   2
      -x x x  + x x x x  + 1), {x , x })
      8 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  - 4x  + x , x , 2x  + 2x  + x , x }), ideal (- 3x  -
                    1     2    4   1    1     2    3   2               1  
      -----------------------------------------------------------------------
                            3        2 2       3     2           2    
      4x x  + x x  + 1, - 8x x  - 16x x  - 8x x  - 4x x x  - 4x 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 :