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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.

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.