Write
$$
P(X)=\sum_{j=0}^np_jX^j,
\qquad
Q(X)=\sum_{j=0}^nq_jX^j,
$$
so there are $N=2(n+1)$ unknown integer coefficients. Let $t$ be the number of integers $r$ with
$$
0\leq r<\frac{(2-\kappa)n}{d}-1.
$$
For each such $r$, impose the linear equation over $K=\mathbb Q(\alpha)$
$$
L_r(\mathbf p,\mathbf q)
=D_r(P+\alpha Q)(\alpha)
=0,
$$
where $D_r$ is the <normalized derivative of a polynomial>. The coefficient of $p_j$ is $\binom jr\alpha^{j-r}$ and that of $q_j$ is $\binom jr\alpha^{j-r+1}$. The local definition of <projective height>, together with $\binom jr\leq2^n$, gives
$$
H_p(L_r)\leq C_0^n
$$
for a constant $C_0$ depending only on $\alpha$.

There are $M=t$ forms over the degree-$d$ field $K$, and
$$
dM<(2-\kappa)n-d<N.
$$
Moreover $N-dM\geq\kappa n+2$. Applying <Siegel lemma> gives a nonzero integral coefficient vector with
$$
\max\{H(P),H(Q)\}
\leq(NC_0^n)^{dM/(N-dM)}
\leq C_1^n,
$$
because 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.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-166/3/b/solution