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

The determinant in the hint is independent of $Y$ and equals the WronskianIt is nonzero because $P,Q$ are linearly independent. Also $\mathrm{deg}W<2n$, and coefficient convolution givesfor a constant $C_0$ depending only on $C$.

Fix $y$, and suppose $F(X,y)$ has multiplicity $r$ at $a=p/q$. Inthe two terms vanish to orders at least $r$ and $r-1$, so $W$ has multiplicity at least $r-1$ at $a$. The primitive polynomial $(qX-p{)}^{r-1}$ therefore divides $W$ in $\mathbb{Z}[X]$ by Gauss lemma for polynomials. Comparing leading coefficients givesIf $r>\delta n+1$, this implies $q^{\delta n}<C_0^n$ and hence $q<C_0^{1/\delta }$. Choosing $q$ larger than this bound proves that $r\le \delta n+1$, uniformly in $y$.

Solved by gpt-5.6-sol high.