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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              1     5                        2   3    
o3 = (map(R,R,{2x  + -x  + x , x , --x  + -x  + x , x }), ideal (3x  + -x x 
                 1   5 2    4   1  10 1   4 2    3   2             1   5 1 2
     ------------------------------------------------------------------------
                 1 3     64 2 2   3   3     2       3   2      1 2      
     + x x  + 1, -x x  + --x x  + -x x  + 2x x x  + -x x x  + --x x x  +
        1 4      5 1 2   25 1 2   4 1 2     1 2 3   5 1 2 3   10 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

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

                    3             5     5         1                         
o6 = (map(R,R,{x  + -x  + x , x , -x  + -x  + x , -x  + x  + x , x }), ideal
                1   7 2    5   1  8 1   7 2    4  5 1    2    3   2         
     ------------------------------------------------------------------------
       2   3               3   3     9 2 2     2       27   3   18   2    
     (x  + -x x  + x x  - x , x x  + -x x  + 3x x x  + --x x  + --x x x  +
       1   7 1 2    1 5    2   1 2   7 1 2     1 2 5   49 1 2    7 1 2 5  
     ------------------------------------------------------------------------
           2    27 4   27 3     9 2 2      3
     3x x x  + ---x  + --x x  + -x x  + x x ), {x , x , x })
       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} | 16807x_1x_2x_5^6-18522x_2^9x_5-243x_2^9+21609x_2^8x_5^2
     {-9}  | 189x_1x_2^2x_5^3-16807x_1x_2x_5^5+441x_1x_2x_5^4+18522x
     {-9}  | 137781x_1x_2^3+12252303x_1x_2^2x_5^2+642978x_1x_2^2x_5+
     {-3}  | 7x_1^2+3x_1x_2+7x_1x_5-7x_2^3                          
     ------------------------------------------------------------------------
                                                                            
     +567x_2^8x_5-16807x_2^7x_5^3-1323x_2^7x_5^2+3087x_2^6x_5^3-7203x_2^5x_5
     _2^9-21609x_2^8x_5-189x_2^8+16807x_2^7x_5^2+882x_2^7x_5-3087x_2^6x_5^2+
     27682574402x_1x_2x_5^5-363182463x_1x_2x_5^4+19059138x_1x_2x_5^3+750141x
                                                                            
     ------------------------------------------------------------------------
                                                                    
     ^4+16807x_2^4x_5^5+7203x_2^2x_5^6+16807x_2x_5^7                
     7203x_2^5x_5^3-16807x_2^4x_5^4+441x_2^4x_5^3+81x_2^3x_5^3-7203x
     _1x_2x_5^2-30507326892x_2^9+35591881374x_2^8x_5+466948881x_2^8-
                                                                    
     ------------------------------------------------------------------------
                                                                             
                                                                             
     _2^2x_5^5+378x_2^2x_5^4-16807x_2x_5^6+441x_2x_5^5                       
     27682574402x_2^7x_5^2-1815912315x_2^7x_5+9529569x_2^7+5084554482x_2^6x_5
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     ^2-66706983x_2^6x_5-1750329x_2^6-11863960458x_2^5x_5^3+155649627x_2^5x_5
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     ^2+4084101x_2^5x_5+321489x_2^5+27682574402x_2^4x_5^4-363182463x_2^4x_5^3
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     +19059138x_2^4x_5^2+750141x_2^4x_5+59049x_2^4+5250987x_2^3x_5^2+413343x_
                                                                             
     ------------------------------------------------------------------------
                                                                         
                                                                         
                                                                         
     2^3x_5+11863960458x_2^2x_5^5-155649627x_2^2x_5^4+20420505x_2^2x_5^3+
                                                                         
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     964467x_2^2x_5^2+27682574402x_2x_5^6-363182463x_2x_5^5+19059138x_2x_5^4+
                                                                             
     ------------------------------------------------------------------------
                    |
                    |
                    |
     750141x_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      9             5      3                      7 2  
o13 = (map(R,R,{-x  + --x  + x , x , -x  + --x  + x , x }), ideal (-x  +
                4 1   10 2    4   1  4 1   10 2    3   2           4 1  
      -----------------------------------------------------------------------
       9                 15 3     27 2 2    27   3   3 2        9   2    
      --x x  + x x  + 1, --x x  + --x x  + ---x x  + -x x x  + --x x x  +
      10 1 2    1 4      16 1 2   20 1 2   100 1 2   4 1 2 3   10 1 2 3  
      -----------------------------------------------------------------------
      5 2        3   2
      -x x x  + --x x x  + x x x x  + 1), {x , x })
      4 1 2 4   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

                      3                   9                        2   3    
o16 = (map(R,R,{3x  + -x  + x , x , 2x  + -x  + x , x }), ideal (4x  + -x x 
                  1   4 2    4   1    1   7 2    3   2             1   4 1 2
      -----------------------------------------------------------------------
                    3     75 2 2   27   3     2       3   2       2      
      + x x  + 1, 6x x  + --x x  + --x x  + 3x x x  + -x x x  + 2x x x  +
         1 4        1 2   14 1 2   28 1 2     1 2 3   4 1 2 3     1 2 4  
      -----------------------------------------------------------------------
      9   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,{2x  - 3x  + x , x , x  + x , x }), ideal (3x  - 3x x  + x x 
                  1     2    4   1   2    3   2             1     1 2    1 4
      -----------------------------------------------------------------------
             2 2       3     2           2        2
      + 1, 2x x  - 3x x  + 2x x x  - 3x x x  + x x x  + x x x x  + 1), {x ,
             1 2     1 2     1 2 3     1 2 3    1 2 4    1 2 3 4         4 
      -----------------------------------------------------------------------
      x })
       3

o19 : Sequence

This symbol is provided by the package NoetherNormalization.

Ways to use noetherNormalization :