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

Writeso there are $N=2(n+1)$ unknown integer coefficients. Let $t$ be the number of integers $r$ withFor each such $r$, impose the linear equation over $K=\mathbb{Q}(\alpha )$where $D_r$ is the normalized derivative of a polynomial. The coefficient of $p_j$ is $\left(\genfrac{}{}{0ex}{}{j}{r}\right){\alpha }^{j-r}$ and that of $q_j$ is $\left(\genfrac{}{}{0ex}{}{j}{r}\right){\alpha }^{j-r+1}$. The local definition of projective height, together with $\left(\genfrac{}{}{0ex}{}{j}{r}\right)\le 2^n$, givesfor a constant $C_0$ depending only on $\alpha$.

There are $M=t$ forms over the degree-$d$ field $K$, andMoreover $N-dM\ge \kappa n+2$. Applying Siegel lemma gives a nonzero integral coefficient vector withbecause the exponent is bounded in terms of $\kappa$ 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.