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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     7             7     9                      14 2   7    
o3 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (--x  + -x x 
               9 1   9 2    4   1  9 1   4 2    3   2            9 1   9 1 2
     ------------------------------------------------------------------------
                 35 3     601 2 2   7   3   5 2       7   2     7 2      
     + x x  + 1, --x x  + ---x x  + -x x  + -x x x  + -x x x  + -x x x  +
        1 4      81 1 2   324 1 2   4 1 2   9 1 2 3   9 1 2 3   9 1 2 4  
     ------------------------------------------------------------------------
     9   2
     -x x x  + x x x x  + 1), {x , x })
     4 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     3                            1                          
o6 = (map(R,R,{-x  + -x  + x , x , x  + 2x  + x , -x  + 7x  + x , x }), ideal
               8 1   4 2    5   1   1     2    4  4 1     2    3   2         
     ------------------------------------------------------------------------
      9 2   3               3  729 3     729 2 2   243 2       243   3  
     (-x  + -x x  + x x  - x , ---x x  + ---x x  + ---x x x  + ---x x  +
      8 1   4 1 2    1 5    2  512 1 2   256 1 2    64 1 2 5   128 1 2  
     ------------------------------------------------------------------------
     81   2     27     2   27 4   27 3     9 2 2      3
     --x x x  + --x x x  + --x  + --x x  + -x x  + x x ), {x , x , x })
     16 1 2 5    8 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} | 9216x_1x_2x_5^6-34992x_2^9x_5-2187x_2^9+23328x_2^8x_5^2+2916x_2
     {-9}  | 216x_1x_2^2x_5^3-2304x_1x_2x_5^5+288x_1x_2x_5^4+8748x_2^9-5832x
     {-9}  | 243x_1x_2^3+2592x_1x_2^2x_5^2+648x_1x_2^2x_5+147456x_1x_2x_5^5-
     {-3}  | 9x_1^2+6x_1x_2+8x_1x_5-8x_2^3                                  
     ------------------------------------------------------------------------
                                                                             
     ^8x_5-10368x_2^7x_5^3-3888x_2^7x_5^2+5184x_2^6x_5^3-6912x_2^5x_5^4+9216x
     _2^8x_5-243x_2^8+2592x_2^7x_5^2+648x_2^7x_5-1296x_2^6x_5^2+1728x_2^5x_5^
     9216x_1x_2x_5^4+2304x_1x_2x_5^3+432x_1x_2x_5^2-559872x_2^9+373248x_2^8x_
                                                                             
     ------------------------------------------------------------------------
                                                                           
     _2^4x_5^5+6144x_2^2x_5^6+8192x_2x_5^7                                 
     3-2304x_2^4x_5^4+288x_2^4x_5^3+144x_2^3x_5^3-1536x_2^2x_5^5+384x_2^2x_
     5+23328x_2^8-165888x_2^7x_5^2-51840x_2^7x_5+1296x_2^7+82944x_2^6x_5^2-
                                                                           
     ------------------------------------------------------------------------
                                                                             
                                                                             
     5^4-2048x_2x_5^6+256x_2x_5^5                                            
     5184x_2^6x_5-648x_2^6-110592x_2^5x_5^3+6912x_2^5x_5^2+864x_2^5x_5+324x_2
                                                                             
     ------------------------------------------------------------------------
                                                                            
                                                                            
                                                                            
     ^5+147456x_2^4x_5^4-9216x_2^4x_5^3+2304x_2^4x_5^2+432x_2^4x_5+162x_2^4+
                                                                            
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     1728x_2^3x_5^2+648x_2^3x_5+98304x_2^2x_5^5-6144x_2^2x_5^4+3840x_2^2x_5^3
                                                                             
     ------------------------------------------------------------------------
                                                                         |
                                                                         |
                                                                         |
     +864x_2^2x_5^2+131072x_2x_5^6-8192x_2x_5^5+2048x_2x_5^4+384x_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
--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     37                   31                      7 2  
o13 = (map(R,R,{-x  + --x  + x , x , 2x  + --x  + x , x }), ideal (-x  +
                5 1   20 2    4   1    1   12 2    3   2           5 1  
      -----------------------------------------------------------------------
      37                 4 3     71 2 2   1147   3   2 2       37   2    
      --x x  + x x  + 1, -x x  + --x x  + ----x x  + -x x x  + --x x x  +
      20 1 2    1 4      5 1 2   15 1 2    240 1 2   5 1 2 3   20 1 2 3  
      -----------------------------------------------------------------------
        2       31   2
      2x x x  + --x x x  + x x x x  + 1), {x , x })
        1 2 4   12 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             9     2                        2   8    
o16 = (map(R,R,{2x  + -x  + x , x , -x  + -x  + x , x }), ideal (3x  + -x x 
                  1   5 2    4   1  4 1   5 2    3   2             1   5 1 2
      -----------------------------------------------------------------------
                  9 3     22 2 2   16   3     2       8   2     9 2      
      + x x  + 1, -x x  + --x x  + --x x  + 2x x x  + -x x x  + -x x x  +
         1 4      2 1 2    5 1 2   25 1 2     1 2 3   5 1 2 3   4 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

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  + x , x , 2x  - x  + x , x }), ideal (x  + x x  + x x  +
                 2    4   1    1    2    3   2            1    1 2    1 4  
      -----------------------------------------------------------------------
           2 2      3      2       2          2
      1, 2x x  - x x  + x x x  + 2x x x  - x x x  + x x x x  + 1), {x , x })
           1 2    1 2    1 2 3     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 :