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eulerMF -- Construct a Koszul matrix factorization with respect to the Jacobian ideal

Description

Constructs a Koszul matrix factorization with respect to the Jacobian ideal. This method works best in characteristic 0. This method utilizes the \( \text{koszulMF} \) function to construct a Koszul matrix factorization with respect to the Jacobian ideal of the polynomial \( f \).

i1 : R = ZZ/101[x,y,z];
 
i2 : f = x^2*y - y^3;
 
i3 : K = eulerMF(f)

      2      2      2
o3 = R  <-- R  <-- R
                    
     0      1      0

o3 : ZZdFactorization
 
i4 : K.dd

          2                       2
o4 = 1 : R  <------------------- R  : 0
               | 34y 2xy     |
               | 34x -x2+3y2 |

          2                       2
     0 : R  <------------------- R  : 1
               | x2-3y2 2xy  |
               | 34x    -34y |

o4 : ZZdFactorizationMap
 
i5 : g = random(4,R)

        4      3       2 2        3      4      3       2           2   
o5 = 24x  - 36x y - 29x y  - 10x*y  - 29y  - 30x z + 19x y*z - 29x*y z -
     ------------------------------------------------------------------------
        3       2 2         2      2 2        3        3      4
     24y z + 19x z  - 8x*y*z  - 38y z  - 22x*z  - 16y*z  + 39z

o5 : R
 
i6 : K' = eulerMF g

      4      4      4
o6 = R  <-- R  <-- R
                    
     0      1      0

o6 : ZZdFactorization
 
i7 : K'.dd

          4                                                                                                                                                                                          4
o7 = 1 : R  <-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- R  : 0
               | -25z 0                                                       -36x3+43x2y-30xy2-15y3+19x2z+43xyz+29y2z-8xz2+25yz2-16z3 -5x3-7x2y+43xy2-10y3+11x2z+38xyz-29y2z+38xz2-8yz2-22z3   |
               | 0    -25z                                                    -25x                                                     25y                                                      |
               | -25y -5x3-7x2y+43xy2-10y3+11x2z+38xyz-29y2z+38xz2-8yz2-22z3  30x3-19x2y+29xy2+24y3-38x2z+16xyz-25y2z-35xz2+48yz2+46z3 0                                                        |
               | -25x 36x3-43x2y+30xy2+15y3-19x2z-43xyz-29y2z+8xz2-25yz2+16z3 0                                                        30x3-19x2y+29xy2+24y3-38x2z+16xyz-25y2z-35xz2+48yz2+46z3 |

          4                                                                                                                                                                                                                                               4
     0 : R  <------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- R  : 1
               | -30x3+19x2y-29xy2-24y3+38x2z-16xyz+25y2z+35xz2-48yz2-46z3 0                                                         -36x3+43x2y-30xy2-15y3+19x2z+43xyz+29y2z-8xz2+25yz2-16z3 -5x3-7x2y+43xy2-10y3+11x2z+38xyz-29y2z+38xz2-8yz2-22z3 |
               | 0                                                         -30x3+19x2y-29xy2-24y3+38x2z-16xyz+25y2z+35xz2-48yz2-46z3 -25x                                                     25y                                                    |
               | -25y                                                      -5x3-7x2y+43xy2-10y3+11x2z+38xyz-29y2z+38xz2-8yz2-22z3    25z                                                      0                                                      |
               | -25x                                                      36x3-43x2y+30xy2+15y3-19x2z-43xyz-29y2z+8xz2-25yz2+16z3   0                                                        25z                                                    |

o7 : ZZdFactorizationMap
 
i8 : potential K' == g

o8 = true

Caveat

Since this method involves computing the jacobian ideal, the characteristic of the underlying field should be sufficiently large with respect to the degree of f.

See also

Ways to use eulerMF:

  • eulerMF(RingElement)

For the programmer

The object eulerMF is a method function.

The source of this document is in MatrixFactorizations/MatrixFactorizationsDOC.m2:5298:0.