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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                  1     7                      9 2         
o3 = (map(R,R,{-x  + x  + x , x , -x  + -x  + x , x }), ideal (-x  + x x  +
               5 1    2    4   1  3 1   5 2    3   2           5 1    1 2  
     ------------------------------------------------------------------------
                4 3     109 2 2   7   3   4 2          2     1 2      
     x x  + 1, --x x  + ---x x  + -x x  + -x x x  + x x x  + -x x x  +
      1 4      15 1 2    75 1 2   5 1 2   5 1 2 3    1 2 3   3 1 2 4  
     ------------------------------------------------------------------------
     7   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

                     1             7     6               1              
o6 = (map(R,R,{9x  + -x  + x , x , -x  + -x  + x , 4x  + -x  + x , x }),
                 1   9 2    5   1  4 1   7 2    4    1   2 2    3   2   
     ------------------------------------------------------------------------
              2   1               3      3        2 2       2       1   3  
     ideal (9x  + -x x  + x x  - x , 729x x  + 27x x  + 243x x x  + -x x  +
              1   9 1 2    1 5    2      1 2      1 2       1 2 5   3 1 2  
     ------------------------------------------------------------------------
         2            2    1  4    1 3     1 2 2      3
     6x x x  + 27x x x  + ---x  + --x x  + -x x  + x x ), {x , x , x })
       1 2 5      1 2 5   729 2   27 2 5   3 2 5    2 5     5   4   3

o6 : Sequence
i7 : transpose gens gb J

o7 = {-10} | x_2^10                                           
     {-10} | 59049x_1x_2x_5^6-39366x_2^9x_5-x_2^9+177147x_2^8x
     {-9}  | 81x_1x_2^2x_5^3-14348907x_1x_2x_5^5+729x_1x_2x_5^
     {-9}  | 81x_1x_2^3+14348907x_1x_2^2x_5^2+1458x_1x_2^2x_5+
     {-3}  | 81x_1^2+x_1x_2+9x_1x_5-9x_2^3                    
     ------------------------------------------------------------------------
                                                                             
     _5^2+9x_2^8x_5-531441x_2^7x_5^3-81x_2^7x_5^2+729x_2^6x_5^3-6561x_2^5x_5^
     4+9565938x_2^9-43046721x_2^8x_5-729x_2^8+129140163x_2^7x_5^2+13122x_2^7x
     33354363399333138x_1x_2x_5^5-847288609443x_1x_2x_5^4+86093442x_1x_2x_5^3
                                                                             
     ------------------------------------------------------------------------
                                                                        
     4+59049x_2^4x_5^5+729x_2^2x_5^6+6561x_2x_5^7                       
     _5-177147x_2^6x_5^2+1594323x_2^5x_5^3-14348907x_2^4x_5^4+729x_2^4x_
     +6561x_1x_2x_5^2-22236242266222092x_2^9+100063090197999414x_2^8x_5+
                                                                        
     ------------------------------------------------------------------------
                                                                            
                                                                            
     5^3+x_2^3x_5^3-177147x_2^2x_5^5+18x_2^2x_5^4-1594323x_2x_5^6+81x_2x_5^5
     2541865828329x_2^8-300189270593998242x_2^7x_5^2-38127987424935x_2^7x_5+
                                                                            
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     387420489x_2^7+411782264189298x_2^6x_5^2-10460353203x_2^6x_5-531441x_2^6
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     -3706040377703682x_2^5x_5^3+94143178827x_2^5x_5^2+4782969x_2^5x_5+729x_2
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     ^5+33354363399333138x_2^4x_5^4-847288609443x_2^4x_5^3+86093442x_2^4x_5^2
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     +6561x_2^4x_5+x_2^4+177147x_2^3x_5^2+27x_2^3x_5+411782264189298x_2^2x_5^
                                                                             
     ------------------------------------------------------------------------
                                                             
                                                             
                                                             
     5-10460353203x_2^2x_5^4+2657205x_2^2x_5^3+243x_2^2x_5^2+
                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     3706040377703682x_2x_5^6-94143178827x_2x_5^5+9565938x_2x_5^4+729x_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

                5     2             2     3                      11 2   2    
o13 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (--x  + -x x 
                6 1   9 2    4   1  3 1   8 2    3   2            6 1   9 1 2
      -----------------------------------------------------------------------
                  5 3     199 2 2    1   3   5 2       2   2     2 2      
      + x x  + 1, -x x  + ---x x  + --x x  + -x x x  + -x x x  + -x x x  +
         1 4      9 1 2   432 1 2   12 1 2   6 1 2 3   9 1 2 3   3 1 2 4  
      -----------------------------------------------------------------------
      3   2
      -x x x  + x x x x  + 1), {x , x })
      8 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     1             6     7                      3 2   1    
o16 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (-x  + -x x 
                2 1   6 2    4   1  5 1   8 2    3   2           2 1   6 1 2
      -----------------------------------------------------------------------
                  3 3     51 2 2    7   3   1 2       1   2     6 2      
      + x x  + 1, -x x  + --x x  + --x x  + -x x x  + -x x x  + -x x x  +
         1 4      5 1 2   80 1 2   48 1 2   2 1 2 3   6 1 2 3   5 1 2 4  
      -----------------------------------------------------------------------
      7   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
--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
--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: 4
--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: 5
--trying with basis element limit: 5
--trying with basis element limit: 20
--warning: no good linear transformation found by noetherNormalization

                                                                   2         
o19 = (map(R,R,{5x  + 23x  + x , x , 5x  + x  + x , x }), ideal (6x  + 23x x 
                  1      2    4   1    1    2    3   2             1      1 2
      -----------------------------------------------------------------------
                     3         2 2        3     2            2       2      
      + x x  + 1, 25x x  + 120x x  + 23x x  + 5x x x  + 23x x x  + 5x x x  +
         1 4         1 2       1 2      1 2     1 2 3      1 2 3     1 2 4  
      -----------------------------------------------------------------------
         2
      x x x  + x x x x  + 1), {x , x })
       1 2 4    1 2 3 4         4   3

o19 : Sequence

This symbol is provided by the package NoetherNormalization.

Ways to use noetherNormalization :