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i1 : -- Example 8c: a genus 8 reducible surface with four components, each generically nonreduced
     S = QQ[x_12,x_13,x_14,x_15,x_16,x_23,x_24,x_25,x_26,x_34,x_35,x_36,x_45,x_46,x_56];

i2 : LNonRedSurf8 = {x_12-x_34, x_12-x_13, x_14-x_45, x_16-x_23, x_24-x_35, x_26-x_36};

i3 : Gr26 = {x_12*x_34-x_13*x_24+x_14*x_23,
     x_12*x_35-x_13*x_25+x_15*x_23,
     x_12*x_36-x_13*x_26+x_16*x_23,
     x_12*x_45-x_14*x_25+x_15*x_24,
     x_12*x_46-x_14*x_26+x_16*x_24,
     x_12*x_56-x_15*x_26+x_16*x_25,
     x_13*x_45-x_14*x_35+x_15*x_34,
     x_13*x_46-x_14*x_36+x_16*x_34,
     x_13*x_56-x_15*x_36+x_16*x_35,
     x_14*x_56-x_15*x_46+x_16*x_45,
     x_23*x_45-x_24*x_35+x_25*x_34,
     x_23*x_46-x_24*x_36+x_26*x_34,
     x_23*x_56-x_25*x_36+x_26*x_35,
     x_24*x_56-x_25*x_46+x_26*x_45,
     x_34*x_56-x_35*x_46+x_36*x_45};

i4 : I = ideal join(LNonRedSurf8,Gr26);

o4 : Ideal of S

i5 : hilbertPolynomial(I,Projective=>false)

       2
o5 = 7i  + 2

o5 : QQ[i]

i6 : radI = radical(I);

o6 : Ideal of S

i7 : I==radI

o7 = false

i8 : pdI = primaryDecomposition(I);

i9 : radpdI = apply(pdI, J -> radical J);

i10 : apply(#pdI, i -> pdI_i==radpdI_i)

o10 = {false, false, false, false}

o10 : List

i11 : apply(pdI, J -> hilbertPolynomial(J,Projective=>false))

         2            2            2             2
o11 = {2i  + 5i + 1, i  + 3i + 2, i  + 5i + 3, 3i  + 9i - 2}

o11 : List

i12 : apply(radpdI, J -> hilbertPolynomial(J,Projective=>false))

        2           1 2   3       1 2   3        2
o12 = {i  + 2i + 1, -i  + -i + 1, -i  + -i + 1, i  + 2i + 1}
                    2     2       2     2

o12 : List

i13 : matrix apply(#pdI, i -> apply(#pdI, j -> dim(radpdI_i+radpdI_j)))

o13 = | 3 0 2 2 |
      | 0 3 1 2 |
      | 2 1 3 2 |
      | 2 2 2 3 |

               4       4
o13 : Matrix ZZ  <-- ZZ

i14 :