Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 24F / b

Table of contents
- Solution b

Solution

Choose a local coordinate

z_{i}

at each

p_{i}

and map

f \in V

to all coefficients of the negative-power terms in its Laurent expansions:

f ⟼ (a_{i, - 1}, \dots, a_{i, - m_{i}})_{i = 1}^{n} \in C^{\sum_{i} m_{i}} .

The kernel consists of globally holomorphic functions on the compact Riemann surface

R

. By the maximum modulus principle, every such function is constant, so the kernel has dimension one. Rank--nullity gives

dim_{C} V \leq 1 + i \sum m_{i} .

For

n = 0

, this simply says

dim V = 1

.

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