Computations - ZR

Reducing the task by solving a linear system

Solve[{zr12 == 
          z2^2*(zr21 + zr22)/z + 
            2  z1*z2*(zr11 + zr12 + zr21 + zr22)/z, 
        zr22 == z2^2*(zr11 + zr12 + zr21 + zr22)/z}, {zr12, 
        zr22}]

zr11 + zr12 + zr21 + zr22 /. %[[1]] // Factor

Denominator[%]

Solve[{% == 0, z1 == z*z2/(1 - z1 - z2)^2, 
        z2 == z/(1 - z1 - z2)}, {z1, z2, z}]

% // N

Thus, the smallest singularity of zr is also the smallest singularity of zr11 resp. zr21.

Solving the system for zr11 and zr21

GroebnerBasis[{zr11*(1 - zr11 - zr21 - z1 - z2)^2 - 
          z*(zr21 + z2), zr21*(1 - zr11 - zr21 - z1 - z2) - z, 
        z1*(1 - z1 - z2)^2 - z*z2, z2*(1 - z1 - z2) - z}, {zr11, z1, 
        z2, zr21}, {zr11, z1, z2}]

%[[1]]/z // Expand

G[zr21_, z_] = %;

This approach takes a lot of computing time, so we do not eliminate z2 at first:

zr21 - z/(1 - zr21^2*(zr21 + z2)/z - zr21 - z2^3/z - z2) // 
      Factor

F[zr21_, z2_, z_] = Numerator[%];

Finding a common zero of the polynomial and its derivative

GroebnerBasis[{F[zr21, z2, z], D[F[zr21, z2, z], zr21], 
        z^2 - z*z2 + z*z2^2 + z2^4}, 
{zr21, z2, z}, {zr21, 
        z2}]

Factor[%[[1]]]

(z^3) (1 - 448 z + 4096 (z^2)) (531441 + 2894400 z + 
          2560000 (z^2))

NSolve[% == 0, z]

So the smallest singularity must be z0=0.107095.

\(z0 = Root[1 - 448\ # + 4096\ #\^2 &, 2]\)

Expanding around the singularity

We determine the values of z1 and z2 at the singularity:

Solve[z0^2 - z0*c2 + z0*c2^2 + c2^4 == 0, c2];

% // N

Since z2 is monotonically increasing with a value of(1/6) ((Sqrt[13]) - 2) = 0.267592 at (1/72) (13 (Sqrt[13]) - 35),we know that c2 = (1/8).

c2 = c2 /. %%[[1]]

c1 = c2^3/z0 // RootReduce

F[zr21, c2, z0]

Solve[% == 0, zr21]

So we know the value of zr21 at z0.

s0 = zr21 /. %[[1]]

N[s0]

We calculate the expansion of zr21 now:

D[G[zr21, z], {zr21, 2}] /. {zr21 -> s0, z -> z0} // 
      RootReduce

D[G[zr21, z], z] /. {zr21 -> s0, z -> z0} // 
      RootReduce

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

N[a]

Then we have zr21~s0 - 
      a (Sqrt[(1 - (z/z0))]).

The expansion of zr11 follows at once:zr11~(((s0^2) (s0 + c2))/z0) - ((a s0(
                            3  s0 + 2  c2))/z0) (Sqrt[(1 - (z/z0))]), where c2 is the value of z2 at z0 .  
            The expansion of sr follows immediately, \(too . \ 
            The\ constant\ coefficient\ is\ \(z0\ \(s0(c2\^2 + c2\ s0 + s0\^2 + z0)\)\)\/\((z0\^2 - 2\ z0\ c1\ c2 - z0\ c2\^2 - c2\^4)\)\),\(the\ coefficient\ of\ \@\((1 - z\/z0)\)\ is\ \(\(-z0\)\ \(a(c2\^2 + s0(3\ s0 + 2\ c2) + z0)\)\)\/\(z0\^2 - z0c2\^2 - \(c1\^2\) c2\^2\)\), \(where\ c1\ is\ the\ value\ of\ s1\ at\ z0\ and\ b\ is\ the\ value\ of\ sr21\ at\ z0 . \ We\ determine\ b\),\(c1\ and\ c2\ first\  - \ 
          we\ use\ Groebner\ bases\ again\ to\ obtain\ exact\ \(\(results\)\(:\)\)\)

z0*s0*(c2^2+c2*s0+s0^2+z0)/(z0^2-2z0*c1*c2-z0*c2^2-c2^4)//RootReduce

% // N

-a*z0(c2^2+s0(3s0+2c2)+z0)/(z0^2-2z0*c1*c2-z0*c2^2-c2^4) //RootReduce

% // N

Solve[125 - 41275200 (x^2) + 99897344 (x^4) == 0, x] // 
      Simplify

Now, we readily obtain the constants in our asymptotics:

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

% // N

Solve[125 - 10318800 (x^2) + 6243584 (x^4) == 0, x] // 
      Simplify

% // N

1/(4*z0) // RootReduce

% // N