Computations - ZZ

Finding a single equation from the system by the Groebner basis approach

GroebnerBasis[{zz11*(1 - zz)^3 - z*zz22*(1 - zz) - 
          2  z*(zz12 + zz22)^2, 
        zz12*(1 - zz)^2 - z*(zz12 + zz22), zz22*(1 - zz) - z, 
        zz - zz11 - 2  zz12 - zz22}, {zz11, zz12, zz22, zz, z}, {zz11, zz12, 
        zz22}]

Collect[%[[1]], zz]

F[zz_, z_] = %;

Finding a common zero of the polynomial and its derivative

GroebnerBasis[{F[zz, z], D[F[zz, z], zz]}, 
{zz, 
        z}, {zz}]

Factor[%[[1]]]

NSolve[% == 0, z]

So the smallest singularity must be z0=0.107969.

\(z0 = 
      Root[1728 - 174839\ # + 1181956\ #\^2 + 2683040\ #\^3 - 45248  #\^4 + 
            2304  #\^5 &, 3]\)

Expanding around the singularity

Solve[F[zz, z0] == 0, zz];

N[%]

s0 = zz /. %%[[1]]

So we have the value of zz at the singularity. We calculate the expansion of zz now:

D[F[zz, z], {zz, 2}] /. {zz -> s0, z -> z0} // 
      RootReduce

D[F[zz, z], z] /. {zz -> s0, z -> z0} // 
      RootReduce

a = Sqrt[2  z0*%/%%] // RootReduce

Then we have zz~s0 - 
      a (Sqrt[(1 - (z/z0))]).

s0 // N

a // N

Now, we readily obtain the constants in our asymptotics:

-a/Gamma[-1/2]*4*Sqrt[Pi] // FullSimplify

% // N

1/(4*z0) // RootReduce

% // N