Solution

The valency theorem states that for a nonconstant analytic map between compact connected Riemann surfaces,

\sum_{z \in f^{- 1} (w)} m_{f} (z) = de g f

for every target point

w

For a rational map

f = p / q

on the Riemann sphere, first cancel common factors; then the degree of a rational map of the Riemann sphere is

de g f = max (de g p, de g q) .

The analytic isomorphisms are exactly the degree-one maps, namely the Möbius transformations.

The required transformation is

h (z) = \frac{z + 1}{z - 1} .

It sends

\infty \leftrightarrow 1

and

0 \leftrightarrow - 1

. Its fixed points satisfy

z^{2} - 2 z - 1 = 0 .

6, 8, 12, 24,

corresponding respectively to vertices, face centres, edge centres, and generic points.

In the displayed

F

, no numerator factor vanishes at

0

or at a fourth root of one. Hence

0, \pm 1, \pm i

are poles of order four. The numerator and denominator have degrees

24

and

20

, so

F (z) \sim z^{4}

at infinity; infinity is also a pole of order four. Thus

F^{- 1} (\infty) = {0, \infty, \pm 1, \pm i}, m_{F} = 4 at each point,

and

de g F = 24 .

Finally let

z

have stabilizer of order

e

. A local coordinate turns its stabilizer action into rotation by

e

th roots of unity. Since

F

is invariant, the first nonconstant term of its local expansion has exponent divisible by

e

, so

m_{F} (z) \geq e

. The orbit has

24/ e

points and therefore contributes at least

24

to the fibre through

F (z)

. The valency theorem and

de g F = 24

F (z) = F (w)

implies that

z

and

w

lie in the same orbit.

Solved by gpt-5.6-sol high.

 Ancestors (10)