makeUnfolding -- Makes the universal homogeneous unfolding of an ideal with positive degree parameters

i1 : L={4,5,7}

o1 = {4, 5, 7}

o1 : List

i2 : I := semigroupIdeal L;

               ZZ
o2 : Ideal of ----[x ..x , x ]
              1009  0   1   3

i3 : (A,unfolding):= makeUnfolding I;

i4 : S=ring I

o4 = S

o4 : PolynomialRing

i5 : fI=res I

      1      3      2
o5 = S  <-- S  <-- S  <-- 0
                           
     0      1      2      3

o5 : ChainComplex

i6 : degs=flatten (gens A/degree)

o6 = {15, 14, 12, 11, 10, 10, 9, 8, 8, 7, 7, 7, 6, 6, 5, 5, 5, 4, 4, 4, 3, 3,
     ------------------------------------------------------------------------
     3, 2, 2, 2, 1, 1, 1}

o6 : List

i7 : n=floor(max degs/2+3)

o7 = 10

i8 : restricted=ideal select(gens A, y-> (degree y)_0<n);

               ZZ
o8 : Ideal of ----[a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a       , a      , a      ]
              1009  {2, 0}   {1, 0}   {0, 0}   {2, 1}   {2, 2}   {1, 1}   {1, 2}   {2, 3}   {0, 1}   {2, 4}   {1, 3}   {0, 2}   {2, 5}   {1, 4}   {2, 6}   {1, 5}   {0, 3}   {2, 7}   {1, 6}   {0, 4}   {2, 8}   {1, 7}   {0, 5}   {2, 9}   {1, 8}   {0, 6}   {2, 10}   {1, 9}   {0, 7}

i9 : SA=ring unfolding

o9 = SA

o9 : PolynomialRing

i10 : runfolding=unfolding%sub(restricted,SA);

               1       3
o10 : Matrix SA  <-- SA

i11 : transpose runfolding

o11 = {-12} | x_0^3-x_1x_3+a_{0, 0}                           |
      {-14} | x_0x_1^2-x_3^2+a_{1, 0}+x_0a_{1, 1}             |
      {-15} | x_1^3-x_0^2x_3+a_{2, 0}+x_0a_{2, 1}+x_1a_{2, 2} |

               3       1
o11 : Matrix SA  <-- SA

i12 : J=flatteningRelations(I,A,runfolding);

                ZZ
o12 : Ideal of ----[a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a       , a      , a      ]
               1009  {2, 0}   {1, 0}   {0, 0}   {2, 1}   {2, 2}   {1, 1}   {1, 2}   {2, 3}   {0, 1}   {2, 4}   {1, 3}   {0, 2}   {2, 5}   {1, 4}   {2, 6}   {1, 5}   {0, 3}   {2, 7}   {1, 6}   {0, 4}   {2, 8}   {1, 7}   {0, 5}   {2, 9}   {1, 8}   {0, 6}   {2, 10}   {1, 9}   {0, 7}

i13 : cJ=decompose J;#cJ

o14 = 1

i15 : ideal prune (A/J)

o15 = ideal ()

                ZZ
o15 : Ideal of ----[a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a      , a       , a      , a      ]
               1009  {1, 1}   {1, 2}   {2, 3}   {0, 1}   {2, 4}   {1, 3}   {0, 2}   {2, 5}   {1, 4}   {2, 6}   {1, 5}   {0, 3}   {2, 7}   {1, 6}   {0, 4}   {2, 8}   {1, 7}   {0, 5}   {2, 9}   {1, 8}   {0, 6}   {2, 10}   {1, 9}   {0, 7}

i16 : family=runfolding%sub(J,SA);

               1       3
o16 : Matrix SA  <-- SA

i17 : betti res ideal family == betti res I

o17 = true

i18 : fiber=ideal sub(family,vars S|random(S^1,S^(numgens A)));

o18 : Ideal of S

i19 : singFiber=radical ideal gens gb (fiber+minors(codim I,jacobian fiber))

o19 = ideal 1

o19 : Ideal of S