 Solution

It is enough to prove the result for $0<\kappa <1$, since a construction for a smaller positive value gives the weaker vanishing requirement for any larger one. Apply part (b) with $\kappa /2$. For large $n$, part (c) gives linearly independent polynomials $P_0,Q_0$, andsatisfies $H(G)\le C_1^n$ and has order at leastat $X=\alpha$ after setting $Y=\alpha$.

Set $\delta =\kappa /(4d)$. By the rational multiplicity bound for a linear auxiliary polynomial, once $q_1$ is sufficiently large, the one-variable polynomial $G(X,y)$ has multiplicity at most $\delta n+1$ at $p_1/q_1$. Consequently there is some integer $j\le \delta n+1$ such thatPut $F=D_j^{X}G$. The normalized derivative of a polynomial preserves integral coefficients and multiplies height by at most $2^n$, so $H(F)\le C_2^n$. Differentiation lowers the vanishing order at $(\alpha ,\alpha )$ by at most $j$; for sufficiently large $n$,Finally write $F=P-YQ$ by replacing the coefficient of $Y$ by its negative. Then $F(p_1/q_1,y)\ne 0$ and all the claimed bounds hold.

Solved by gpt-5.6-sol high.