There are several ways (Hilbert's Nullstellensatz, model theory, transcendence bases etc.) to prove the following (amazing!) result:
If $f_1,...,f_r$ is a system of polynomials in $n$ variables with integral coefficients, then it has a solution with coordinates in $\mathbb{C}$ if and only if it has solutions with coordinates in $\overline{\mathbb{F}_p}$ for almost all primes $p$.
Question: What are interesting, explicit examples of the implication which yields solutions over finite fields out of a complex solution? Is there a system of polynomials, where the primes $p$ such that there is a solution over $\overline{\mathbb{F}_p}$ are not known, and their existence is only known by the abstract result above? I am not interested in polynomials which somehow artifically encode some undecidable statements of ZFC ;).