Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 166 / 3 / b

Solution

Write

P (X) = \sum_{j = 0}^{n} p_{j} X^{j}, Q (X) = \sum_{j = 0}^{n} q_{j} X^{j},

so there are

N = 2 (n + 1)

t

r

with

0 \leq r < \frac{( 2 - κ ) n}{d} - 1.

For each such

r

, impose the linear equation over

K = Q (α)

L_{r} (p, q) = D_{r} (P + α Q) (α) = 0,

where

D_{r}

p_{j}

(r j) α^{j - r}

and that of

q_{j}

(r j) α^{j - r + 1}

. The local definition of projective height, together with

(r j) \leq 2^{n}

, gives

H_{p} (L_{r}) \leq C_{0}^{n}

for a constant

C_{0}

depending only on

α

There are

M = t

forms over the degree-

d

K

, and

d M < (2 - κ) n - d < N .

Moreover

N - d M \geq κn + 2

max {H (P), H (Q)} \leq (N C_{0}^{n})^{d M / (N - d M)} \leq C_{1}^{n},

because the exponent is bounded in terms of

κ

and every fixed power of

n

is at most exponential in

n

. These

P, Q

have all the required vanishing normalized derivatives.

Solved by gpt-5.6-sol high.

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