Computations - SS

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

GroebnerBasis[{ss11*(1 - ss22) - z, ss12*(1 - ss12 - ss22) - z, 
        ss22*(1 - ss) - z, ss - ss11 - 2  ss12 - ss22}, {ss11, ss12, 
        ss22, ss, z}, {ss11, ss12, ss22}]

Collect[%[[1]], ss]

F[ss_, z_] = %;

Finding a common zero of the polynomial and its derivative

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

Factor[%[[1]]]

NSolve[% == 0, z]

So the smallest singularity must be z0=0.08738321068182951`.

\(z0 = 
      Root[\(-400\) + 4048  # + 6857  #\^2 - 14848  #\^3 + 65536\ #\^4 &, 
        2]\)

Expanding around the singularity

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

N[%]

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

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

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

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

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

Then we have ss~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