If $P=\sum_{\bf i} a_{\bf i}x^{\bf i}\in {\mathbb Z}[x_1,\dots,x_d]$, let $|P|=\sum_{\bf i}|a_{\bf i}|x^{\bf i}$ and $h(P)=|P|(2,\dots,2)$, so that there is only a finite number of $P$ with $h(P)\leq N$ when $d$ varies. What size would be the smallest $N$ for which:

$\bullet$ One does not know the integral solutions of $P(x)=0$.

$\bullet$ There is a deterministic algorithm to find the integral solutions of $P(x)=0$, but the involved bounds are too big.

$\bullet$ One knows the integral solutions of $P(x)=0$ but not its rational solutions.

$\bullet$ One knows the integral solutions of $P(x)=0$ to be undecidable.

$\bullet$ One knows the rational solutions of $P(x)=0$ to be undecidable.

If $P=\sum_{\bf i} a_{\bf i}x^{\bf i}\in {\mathbb Z}[x_1,\dots,x_d]$, let $|P|=\sum_{\bf i}|a_{\bf i}|x^{\bf i}$ and $h(P)=|P|(2,\dots,2)$, so that there is only a finite number of $P$ with $h(P)\leq N$ when $d$ varies. What size would be the smallest $N$ for which:

• One does not know the integral solutions of $P(x)=0$.

• There is a deterministic algorithm to find the integral solutions of $P(x)=0$, but the involved bounds are too big.

• One knows the integral solutions of $P(x)=0$ but not its rational solutions.

• One knows the integral solutions of $P(x)=0$ to be undecidable.

• One knows the rational solutions of $P(x)=0$ to be undecidable.