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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

               1     1                  8                      5 2   1      
o3 = (map(R,R,{-x  + -x  + x , x , x  + -x  + x , x }), ideal (-x  + -x x  +
               4 1   9 2    4   1   1   5 2    3   2           4 1   9 1 2  
     ------------------------------------------------------------------------
               1 3     23 2 2    8   3   1 2       1   2      2       8   2
     x x  + 1, -x x  + --x x  + --x x  + -x x x  + -x x x  + x x x  + -x x x 
      1 4      4 1 2   45 1 2   45 1 2   4 1 2 3   9 1 2 3    1 2 4   5 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

                                         1         4     9              
o6 = (map(R,R,{5x  + 3x  + x , x , 2x  + -x  + x , -x  + -x  + x , x }),
                 1     2    5   1    1   5 2    4  5 1   2 2    3   2   
     ------------------------------------------------------------------------
              2                   3      3         2 2      2             3  
     ideal (5x  + 3x x  + x x  - x , 125x x  + 225x x  + 75x x x  + 135x x  +
              1     1 2    1 5    2      1 2       1 2      1 2 5       1 2  
     ------------------------------------------------------------------------
          2            2      4      3       2 2      3
     90x x x  + 15x x x  + 27x  + 27x x  + 9x x  + x x ), {x , x , x })
        1 2 5      1 2 5      2      2 5     2 5    2 5     5   4   3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10                                                        
     {-10} | 5x_1x_2x_5^6-1350x_2^9x_5-1215x_2^9+225x_2^8x_5^2+405x_2^8x_5-
     {-9}  | 135x_1x_2^2x_5^3-25x_1x_2x_5^5+45x_1x_2x_5^4+6750x_2^9-1125x_2
     {-9}  | 98415x_1x_2^3+18225x_1x_2^2x_5^2+65610x_1x_2^2x_5+1250x_1x_2x_
     {-3}  | 5x_1^2+3x_1x_2+x_1x_5-x_2^3                                   
     ------------------------------------------------------------------------
                                                                             
     25x_2^7x_5^3-135x_2^7x_5^2+45x_2^6x_5^3-15x_2^5x_5^4+5x_2^4x_5^5+3x_2^2x
     ^8x_5-675x_2^8+125x_2^7x_5^2+450x_2^7x_5-225x_2^6x_5^2+75x_2^5x_5^3-25x_
     5^5-1125x_1x_2x_5^4+4050x_1x_2x_5^3+10935x_1x_2x_5^2-337500x_2^9+56250x_
                                                                             
     ------------------------------------------------------------------------
                                                                            
     _5^6+x_2x_5^7                                                          
     2^4x_5^4+45x_2^4x_5^3+81x_2^3x_5^3-15x_2^2x_5^5+54x_2^2x_5^4-5x_2x_5^6+
     2^8x_5+50625x_2^8-6250x_2^7x_5^2-28125x_2^7x_5+10125x_2^7+11250x_2^6x_5
                                                                            
     ------------------------------------------------------------------------
                                                                            
                                                                            
     9x_2x_5^5                                                              
     ^2-10125x_2^6x_5-18225x_2^6-3750x_2^5x_5^3+3375x_2^5x_5^2+6075x_2^5x_5+
                                                                            
     ------------------------------------------------------------------------
                                                                           
                                                                           
                                                                           
     32805x_2^5+1250x_2^4x_5^4-1125x_2^4x_5^3+4050x_2^4x_5^2+10935x_2^4x_5+
                                                                           
     ------------------------------------------------------------------------
                                                                          
                                                                          
                                                                          
     59049x_2^4+10935x_2^3x_5^2+59049x_2^3x_5+750x_2^2x_5^5-675x_2^2x_5^4+
                                                                          
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     6075x_2^2x_5^3+19683x_2^2x_5^2+250x_2x_5^6-225x_2x_5^5+810x_2x_5^4+2187x
                                                                             
     ------------------------------------------------------------------------
             |
             |
             |
     _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     1                  5                      12 2   1    
o13 = (map(R,R,{-x  + -x  + x , x , x  + -x  + x , x }), ideal (--x  + -x x 
                5 1   4 2    4   1   1   4 2    3   2            5 1   4 1 2
      -----------------------------------------------------------------------
                  7 3       2 2    5   3   7 2       1   2      2      
      + x x  + 1, -x x  + 2x x  + --x x  + -x x x  + -x x x  + x x x  +
         1 4      5 1 2     1 2   16 1 2   5 1 2 3   4 1 2 3    1 2 4  
      -----------------------------------------------------------------------
      5   2
      -x x x  + x x x x  + 1), {x , x })
      4 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

                10     3                   3                      17 2  
o16 = (map(R,R,{--x  + -x  + x , x , 4x  + -x  + x , x }), ideal (--x  +
                 7 1   5 2    4   1    1   2 2    3   2            7 1  
      -----------------------------------------------------------------------
      3                 40 3     159 2 2    9   3   10 2       3   2    
      -x x  + x x  + 1, --x x  + ---x x  + --x x  + --x x x  + -x x x  +
      5 1 2    1 4       7 1 2    35 1 2   10 1 2    7 1 2 3   5 1 2 3  
      -----------------------------------------------------------------------
        2       3   2
      4x x x  + -x x x  + x x x x  + 1), {x , x })
        1 2 4   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
--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

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

o19 : Sequence

This symbol is provided by the package NoetherNormalization.

Ways to use noetherNormalization :