Computations - SZ

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

GroebnerBasis[{sz11*(1 - sz21 - sz22)^2 - z*sz22, 
        sz12*(1 - sz21 - sz22) - z, 
        sz21*(1 - sz)^2 - z*(sz12 + sz22), sz22*(1 - sz) - z, 
        sz - sz11 - sz12 - sz21 - sz22}, {sz11, sz12, sz21, sz22, sz}, {sz11, 
        sz12, sz21, sz}]

%[[1]]/z // Expand

F[sz22_, z_] = %;

Finding a common zero of the polynomial and its derivative

GroebnerBasis[{F[sz22, z], 
        D[F[sz22, z], sz22]}, 
{sz22, 
        z}, {sz22}]

Factor[%[[1]]]

NSolve[% == 0, z]

So the smallest singularity must be z0=0.0982673.

\(z0 = 
      Root[\(-387420489\) + 3451610880\ # + 5017360096\ #\^2 - 
            221833728\ #\^3 + 5038848\ #\^4 &, 2]\)

Expanding around the singularity

Solve[F[sz22, z0] == 0, sz22];

N[%]

s0 = sz22 /. %%[[1]]

So we have the value of sz22 at the singularity. We calculate the expansion of sz22 now:

D[F[sz22, z], {sz22, 2}] /. {sz22 -> s0, z -> z0} // 
      RootReduce

D[F[sz22, z], z] /. {sz22 -> s0, z -> z0} // 
      RootReduce

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

Then we have sz22~s0 - 
      a (Sqrt[(1 - (z/z0))]).

s0 // N

a // N

The expansion of sz follows at once :  
        sz22~1 - 
          (z0/s0) - ((a z0)/(s0^2)) (Sqrt[(1 - 
                      (z/z0))]).

1 - z0/s0 // N

a*z0/s0^2 // RootReduce

% // N

Now, we readily obtain the constants in our asymptotics:

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

% // N

1/(4*z0) // RootReduce

% // N