-- -*- M2-comint -*- {* hash: 684437939 *}

i1 : A = ZZ/101[x,y,z]

o1 = A

o1 : PolynomialRing

i2 : F = x^4 - y*z*(1-x)^2 - z - y^3

      4    2       3
o2 = x  - x y*z - y  + 2x*y*z - y*z - z

o2 : A

i3 : I = ideal(F,diff(x,F),diff(y,F),diff(z,F))

             4    2       3                       3                     2   
o3 = ideal (x  - x y*z - y  + 2x*y*z - y*z - z, 4x  - 2x*y*z + 2y*z, - x z -
     ------------------------------------------------------------------------
       2                2
     3y  + 2x*z - z, - x y + 2x*y - y - 1)

o3 : Ideal of A

i4 : 1 % I

o4 = 0

o4 : A

i5 : g = gens I

o5 = | x4-x2yz-y3+2xyz-yz-z 4x3-2xyz+2yz -x2z-3y2+2xz-z -x2y+2xy-y-1 |

             1       4
o5 : Matrix A  <--- A

i6 : f = matrix{{1_A}}

o6 = | 1 |

             1       1
o6 : Matrix A  <--- A

i7 : h = f // g

o7 = {4} | 44x2y2+43x3z+42x2yz-30x3-26x2y-11xy2+21x2z-7xyz+30x2+15xy-36y2+42x
     {3} | -11x3y2-36x4z+40x3yz-43x4-44x3y+9x2y2+20x3z+27x2yz+43x3-29x2y+22xy
     {3} | 19x2y3-48x3yz-14x2y2z+10x3y-25x2y2-30xy3-7x2yz+36xy2z-10x2y-5xy2+1
     {3} | -41x2y2z-24x3z2-7x2yz2+25x3y+5x3z+38x2yz+45xy2z+18x2z2+39xyz2+x2y-
     ------------------------------------------------------------------------
     z+37yz-2x+2y+9z+31                                    |
     2+40x2z+40xyz-50x2+50xy-20y2-28xz-6yz-33x-19y-47z     |
     2y3-14xyz-46y2z-2x2-33xy+33y2-19xz-3yz+35x-44y+24z-34 |
     25x2z+35xyz-16y2z+25xz2-3yz2-2xy+8yz-33z2+y+3z-1      |

             4       1
o7 : Matrix A  <--- A

i8 : g * (f//g)

o8 = | 1 |

             1       1
o8 : Matrix A  <--- A

i9 : 1 // (gens I)

o9 = {4} | 44x2y2+43x3z+42x2yz-30x3-26x2y-11xy2+21x2z-7xyz+30x2+15xy-36y2+42x
     {3} | -11x3y2-36x4z+40x3yz-43x4-44x3y+9x2y2+20x3z+27x2yz+43x3-29x2y+22xy
     {3} | 19x2y3-48x3yz-14x2y2z+10x3y-25x2y2-30xy3-7x2yz+36xy2z-10x2y-5xy2+1
     {3} | -41x2y2z-24x3z2-7x2yz2+25x3y+5x3z+38x2yz+45xy2z+18x2z2+39xyz2+x2y-
     ------------------------------------------------------------------------
     z+37yz-2x+2y+9z+31                                    |
     2+40x2z+40xyz-50x2+50xy-20y2-28xz-6yz-33x-19y-47z     |
     2y3-14xyz-46y2z-2x2-33xy+33y2-19xz-3yz+35x-44y+24z-34 |
     25x2z+35xyz-16y2z+25xz2-3yz2-2xy+8yz-33z2+y+3z-1      |

             4       1
o9 : Matrix A  <--- A

i10 : M = matrix{{1,x,y},{x,0,y},{1,2,3}}

o10 = | 1 x y |
      | x 0 y |
      | 1 2 3 |

              3       3
o10 : Matrix A  <--- A

i11 : M = substitute(M, frac A)

o11 = | 1 x y |
      | x 0 y |
      | 1 2 3 |

                     3              3
o11 : Matrix (frac A)  <--- (frac A)

i12 : det M

          2
o12 = - 3x  + 3x*y - 2y

o12 : frac(A)

i13 : Minv = id_(target M) // M

o13 = {1} | -33y/(x2-xy-33y)    (x+33y)/(x2-xy-33y)   -34xy/(x2-xy-33y)      
      {1} | (x-34y)/(x2-xy-33y) (34y-1)/(x2-xy-33y)   (-34xy+34y)/(x2-xy-33y)
      {1} | 33x/(x2-xy-33y)     (-34x-33)/(x2-xy-33y) 34x2/(x2-xy-33y)       
      -----------------------------------------------------------------------
      |
      |
      |

                     3              3
o13 : Matrix (frac A)  <--- (frac A)

i14 : M * Minv

o14 = | 1 0 0 |
      | 0 1 0 |
      | 0 0 1 |

                     3              3
o14 : Matrix (frac A)  <--- (frac A)

i15 :