Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 125 / 3 / b

Table of contents
- Solution b

Solution

Write

X = x \circ ϕ

. Since

X (- P) = X (P)

, it is an even rational function and therefore belongs to

k (x)

; write

X = p (x) / q (x)

with coprime polynomials

p, q

. The pullback of a nonzero invariant differential is invariant, so for some

c \in k^{\times}

,

ϕ^{*} (\frac{d X}{2 Y}) = c \frac{d x}{2 y} .

Since

d X = \frac{p ^{'} q - p q ^{'}}{q ^{2}}, d x,

we obtain

Y = \frac{p ^{'} q - p q ^{'}}{c q ^{2}}, y .

Thus

ϕ (x, y) = (\frac{p ( x )}{q ( x )}, \frac{p ^{'} ( x ) q ( x ) - p ( x ) q ^{'} ( x )}{c q ( x ) ^{2}} y) .

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