next | previous | forward | backward | up | top | index | toc | Macaulay2 web site
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     3                        2   5      
o3 = (map(R,R,{4x  + -x  + x , x , -x  + -x  + x , x }), ideal (5x  + -x x  +
                 1   9 2    4   1  2 1   8 2    3   2             1   9 1 2  
     ------------------------------------------------------------------------
                  3     31 2 2    5   3     2       5   2     7 2      
     x x  + 1, 14x x  + --x x  + --x x  + 4x x x  + -x x x  + -x x x  +
      1 4         1 2    9 1 2   24 1 2     1 2 3   9 1 2 3   2 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

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     10             2     4         3     3              
o6 = (map(R,R,{--x  + --x  + x , x , -x  + -x  + x , -x  + -x  + x , x }),
               10 1    7 2    5   1  3 1   7 2    4  8 1   4 2    3   2   
     ------------------------------------------------------------------------
             3 2   10               3   27  3     27 2 2    27 2       90   3
     ideal (--x  + --x x  + x x  - x , ----x x  + --x x  + ---x x x  + --x x 
            10 1    7 1 2    1 5    2  1000 1 2   70 1 2   100 1 2 5   49 1 2
     ------------------------------------------------------------------------
       18   2      9     2   1000 4   300 3     30 2 2      3
     + --x x x  + --x x x  + ----x  + ---x x  + --x x  + x x ), {x , x , x })
        7 1 2 5   10 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} | 504210x_1x_2x_5^6-1852200x_2^9x_5-3000000x_2^9+648270x_2
     {-9}  | 21000000x_1x_2^2x_5^3-4537890x_1x_2x_5^5+14700000x_1x_2x
     {-9}  | 10500000000000000x_1x_2^3+2268945000000000x_1x_2^2x_5^2+
     {-3}  | 21x_1^2+100x_1x_2+70x_1x_5-70x_2^3                      
     ------------------------------------------------------------------------
                                                             
     ^8x_5^2+2100000x_2^8x_5-151263x_2^7x_5^3-1470000x_2^7x_5
     _5^4+16669800x_2^9-5834430x_2^8x_5-6300000x_2^8+1361367x
     14700000000000000x_1x_2^2x_5+100902983695290x_1x_2x_5^5-
                                                             
     ------------------------------------------------------------------------
                                                                             
     ^2+1029000x_2^6x_5^3-720300x_2^5x_5^4+504210x_2^4x_5^5+2401000x_2^2x_5^6
     _2^7x_5^2+8820000x_2^7x_5-9261000x_2^6x_5^2+6482700x_2^5x_5^3-4537890x_2
     163432108350000x_1x_2x_5^4+1058841000000000x_1x_2x_5^3+5145000000000000x
                                                                             
     ------------------------------------------------------------------------
                                                             
     +1680700x_2x_5^7                                        
     ^4x_5^4+14700000x_2^4x_5^3+100000000x_2^3x_5^3-21609000x
     _1x_2x_5^2-370664021737800x_2^9+129732407608230x_2^8x_5+
                                                             
     ------------------------------------------------------------------------
                                                                           
                                                                           
     _2^2x_5^5+140000000x_2^2x_5^4-15126300x_2x_5^6+49000000x_2x_5^5       
     210126996450000x_2^8-30270895108587x_2^7x_5^2-245148162525000x_2^7x_5+
                                                                           
     ------------------------------------------------------------------------
                                                                            
                                                                            
                                                                            
     158826150000000x_2^7+205924456521000x_2^6x_5^2-333534915000000x_2^6x_5-
                                                                            
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     1080450000000000x_2^6-144147119564700x_2^5x_5^3+233474440500000x_2^5x_5^
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     2+756315000000000x_2^5x_5+7350000000000000x_2^5+100902983695290x_2^4x_5^
                                                                             
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     4-163432108350000x_2^4x_5^3+1058841000000000x_2^4x_5^2+5145000000000000x
                                                                             
     ------------------------------------------------------------------------
                                                                
                                                                
                                                                
     _2^4x_5+50000000000000000x_2^4+10804500000000000x_2^3x_5^2+
                                                                
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     105000000000000000x_2^3x_5+480490398549000x_2^2x_5^5-778248135000000x_2^
                                                                             
     ------------------------------------------------------------------------
                                                                    
                                                                    
                                                                    
     2x_5^4+12605250000000000x_2^2x_5^3+73500000000000000x_2^2x_5^2+
                                                                    
     ------------------------------------------------------------------------
                                                                             
                                                                             
                                                                             
     336343278984300x_2x_5^6-544773694500000x_2x_5^5+3529470000000000x_2x_5^4
                                                                             
     ------------------------------------------------------------------------
                                |
                                |
                                |
     +17150000000000000x_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

                10                   7     1                      13 2  
o13 = (map(R,R,{--x  + 5x  + x , x , -x  + -x  + x , x }), ideal (--x  +
                 3 1     2    4   1  2 1   6 2    3   2            3 1  
      -----------------------------------------------------------------------
                        35 3     325 2 2   5   3   10 2           2    
      5x x  + x x  + 1, --x x  + ---x x  + -x x  + --x x x  + 5x x x  +
        1 2    1 4       3 1 2    18 1 2   6 1 2    3 1 2 3     1 2 3  
      -----------------------------------------------------------------------
      7 2       1   2
      -x x x  + -x x x  + x x x x  + 1), {x , x })
      2 1 2 4   6 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     8             1     3                      5 2   8    
o16 = (map(R,R,{-x  + -x  + x , x , -x  + -x  + x , x }), ideal (-x  + -x x 
                2 1   9 2    4   1  6 1   7 2    3   2           2 1   9 1 2
      -----------------------------------------------------------------------
                  1 3     299 2 2    8   3   3 2       8   2     1 2      
      + x x  + 1, -x x  + ---x x  + --x x  + -x x x  + -x x x  + -x x x  +
         1 4      4 1 2   378 1 2   21 1 2   2 1 2 3   9 1 2 3   6 1 2 4  
      -----------------------------------------------------------------------
      3   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,{- 4x  - x  + x , x , - 4x  + 2x  + x , x }), ideal (- 3x  -
                    1    2    4   1      1     2    3   2               1  
      -----------------------------------------------------------------------
                          3       2 2       3     2          2       2      
      x x  + x x  + 1, 16x x  - 4x x  - 2x x  - 4x x x  - x x x  - 4x x x  +
       1 2    1 4         1 2     1 2     1 2     1 2 3    1 2 3     1 2 4  
      -----------------------------------------------------------------------
          2
      2x 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 :