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The naive height of a polynomial is the largest absolute value of its coefficients. For a primitive polynomial in $\mathbb{Z}[X_1,\dots ,X_n]$, this equals the projective height of its coefficient vector.

For a linear form $L=a_1{X}_1+\cdots +a_N{X}_N$ over a number field $K$, define its height to be the projective height of its coefficient vector:The product formula makes this independent of multiplying $L$ by a nonzero scalar.

The Siegel lemma says the following. Let $D=[K:\mathbb{Q}]$, let $L_1,\dots ,L_M$ be linear forms in $N$ variables with $DM<N$, and suppose $H_p(L_i)\le \mathcal{H}$ for every $i$, where $\mathcal{H}\ge 1$. Then there is a nonzero $\mathbf{x}\in {\mathbb{Z}}^N$ annihilated by all the forms and satisfying

Solved by gpt-5.6-sol high.

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.

Let $K$ be a number field of degree $D$, and let $L_1,\dots ,L_M$ be linear forms in $N>DM$ variables with coefficients in $K$ and projective height at most $\mathcal{H}\ge 1$. There is a nonzero $\mathbf{x}\in {\mathbb{Z}}^N$ satisfying every $L_j(\mathbf{x})=0$ andFor $K=\mathbb{Q}$, this follows by applying the pigeonhole principle to the images of the integer box $\{0,\dots ,Y{\}}^N$ under $(L_1,\dots ,L_M)$. Expanding coefficients in a rational basis of $K$ gives the factor $D$ in the general count.