As I mentioned in the other thread, <a href="https://en.wikipedia.org/wiki/Diophantine_set#Matiyasevich.27s_theorem">Matiyasevich's theorem</a> implies that it is undecidable whether a system of Diophantine equations over $\mathbb{Z}$ has a solution (Hilbert's 10th Problem).  I have to mention some related results here: if $\mathbb{Z}$ is replaced by $\mathbb{F}_p[t]$ then the problem is still not decidable, if replaced by $\mathbb{R}, \mathbb{C}, \mathbb{Q}_p$ then the problem is decidable, and if replaced by $\mathbb{Q}$ or $\mathbb{F}_p((t))$ the answer is not known!  (<a href="http://www-math.mit.edu/~poonen/papers/aws2003.pdf">Reference</a>.)  I believe it is not even known whether the answer is yes for some number fields but no for others (which would be truly bizarre).