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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

               6     5                  1                      13 2   5      
o3 = (map(R,R,{-x  + -x  + x , x , x  + -x  + x , x }), ideal (--x  + -x x  +
               7 1   9 2    4   1   1   7 2    3   2            7 1   9 1 2  
     ------------------------------------------------------------------------
               6 3     299 2 2    5   3   6 2       5   2      2      
     x x  + 1, -x x  + ---x x  + --x x  + -x x x  + -x x x  + x x x  +
      1 4      7 1 2   441 1 2   63 1 2   7 1 2 3   9 1 2 3    1 2 4  
     ------------------------------------------------------------------------
     1   2
     -x x x  + x x x x  + 1), {x , x })
     7 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             1               9     10              
o6 = (map(R,R,{-x  + -x  + x , x , -x  + 2x  + x , -x  + --x  + x , x }),
               3 1   4 2    5   1  5 1     2    4  5 1    7 2    3   2   
     ------------------------------------------------------------------------
            1 2   7               3   1 3      7 2 2   1 2       49   3  
     ideal (-x  + -x x  + x x  - x , --x x  + --x x  + -x x x  + --x x  +
            3 1   4 1 2    1 5    2  27 1 2   12 1 2   3 1 2 5   16 1 2  
     ------------------------------------------------------------------------
     7   2          2   343 4   147 3     21 2 2      3
     -x x x  + x x x  + ---x  + ---x x  + --x x  + x x ), {x , x , x })
     2 1 2 5    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} | 3072x_1x_2x_5^6-18816x_2^9x_5-50421x_2^9+5376x_2^8x_5^2+28812x
     {-9}  | 28812x_1x_2^2x_5^3-3072x_1x_2x_5^5+16464x_1x_2x_5^4+18816x_2^9
     {-9}  | 71183762748x_1x_2^3+7589772288x_1x_2^2x_5^2+81352871712x_1x_2^
     {-3}  | 4x_1^2+21x_1x_2+12x_1x_5-12x_2^3                              
     ------------------------------------------------------------------------
                                                                         
     _2^8x_5-1024x_2^7x_5^3-16464x_2^7x_5^2+9408x_2^6x_5^3-5376x_2^5x_5^4
     -5376x_2^8x_5-9604x_2^8+1024x_2^7x_5^2+10976x_2^7x_5-9408x_2^6x_5^2+
     2x_5+100663296x_1x_2x_5^5-269746176x_1x_2x_5^4+2891341824x_1x_2x_5^3
                                                                         
     ------------------------------------------------------------------------
                                                                             
     +3072x_2^4x_5^5+16128x_2^2x_5^6+9216x_2x_5^7                            
     5376x_2^5x_5^3-3072x_2^4x_5^4+16464x_2^4x_5^3+151263x_2^3x_5^3-16128x_2^
     +23243677632x_1x_2x_5^2-616562688x_2^9+176160768x_2^8x_5+472055808x_2^8-
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
     2x_5^5+172872x_2^2x_5^4-9216x_2x_5^6+49392x_2x_5^5                      
     33554432x_2^7x_5^2-449576960x_2^7x_5+481890304x_2^7+308281344x_2^6x_5^2-
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     826097664x_2^6x_5-4427367168x_2^6-176160768x_2^5x_5^3+472055808x_2^5x_5^
                                                                             
     ------------------------------------------------------------------------
                                                                            
                                                                            
                                                                            
     2+2529924096x_2^5x_5+40676435856x_2^5+100663296x_2^4x_5^4-269746176x_2^
                                                                            
     ------------------------------------------------------------------------
                                                                       
                                                                       
                                                                       
     4x_5^3+2891341824x_2^4x_5^2+23243677632x_2^4x_5+373714754427x_2^4+
                                                                       
     ------------------------------------------------------------------------
                                                                    
                                                                    
                                                                    
     39846304512x_2^3x_5^2+640653864732x_2^3x_5+528482304x_2^2x_5^5-
                                                                    
     ------------------------------------------------------------------------
                                                                       
                                                                       
                                                                       
     1416167424x_2^2x_5^4+37948861440x_2^2x_5^3+366087922704x_2^2x_5^2+
                                                                       
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     301989888x_2x_5^6-809238528x_2x_5^5+8674025472x_2x_5^4+69731032896x_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

                7                  2                           12 2         
o13 = (map(R,R,{-x  + x  + x , x , -x  + x  + x , x }), ideal (--x  + x x  +
                5 1    2    4   1  3 1    2    3   2            5 1    1 2  
      -----------------------------------------------------------------------
                14 3     31 2 2      3   7 2          2     2 2          2
      x x  + 1, --x x  + --x x  + x x  + -x x x  + x x x  + -x x x  + x x x 
       1 4      15 1 2   15 1 2    1 2   5 1 2 3    1 2 3   3 1 2 4    1 2 4
      -----------------------------------------------------------------------
      + x x x x  + 1), {x , x })
         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     5             8     1                      17 2  
o16 = (map(R,R,{--x  + -x  + x , x , -x  + -x  + x , x }), ideal (--x  +
                 7 1   9 2    4   1  3 1   3 2    3   2            7 1  
      -----------------------------------------------------------------------
      5                 80 3     370 2 2    5   3   10 2       5   2    
      -x x  + x x  + 1, --x x  + ---x x  + --x x  + --x x x  + -x x x  +
      9 1 2    1 4      21 1 2   189 1 2   27 1 2    7 1 2 3   9 1 2 3  
      -----------------------------------------------------------------------
      8 2       1   2
      -x x x  + -x x x  + x x x x  + 1), {x , x })
      3 1 2 4   3 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,{2x  + x  + x , x , x  + 2x  + x , x }), ideal (3x  + x x  +
                  1    2    4   1   1     2    3   2             1    1 2  
      -----------------------------------------------------------------------
                  3       2 2       3     2          2      2           2
      x x  + 1, 2x x  + 5x x  + 2x x  + 2x x x  + x x x  + x x x  + 2x x x  +
       1 4        1 2     1 2     1 2     1 2 3    1 2 3    1 2 4     1 2 4  
      -----------------------------------------------------------------------
      x x x x  + 1), {x , x })
       1 2 3 4         4   3

o19 : Sequence

This symbol is provided by the package NoetherNormalization.

Ways to use noetherNormalization :