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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 ;).

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    Indicate the nature of the coefficients of the f_i (e.g., they can't be random complex numbers or it wouldn't make sense to speak of solns in char. p) and what "almost all" means.2011-04-24
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    I assume the $f_i$ have integer coefficients and "almost all" means "all but finitely many." But I am not sure what to make of the question. Even when $r = 1$ how are you supposed to produce a root of an integer polynomial over $\overline{\mathbb{F}_p}$ out of a complex root? I don't think you can say anything more than "the two fields are both algebraically closed, therefore this polynomial admits a root over both of them."2011-04-24
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    Qiaochu: Yes, I too presumed that Martin meant to specify those condition just as you wrote them. But I still think he should include them in his question.2011-04-25
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    I can't give an example, but I can give a very interesting application of the "if" direction (which I think can also be proved using complex analysis): if $f: \mathbb{C}^n \to \mathbb{C}^n$ is an injective polynomial map (with integer coefficients), then $f$ is surjective.2011-04-25
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    @KCd: I've edited the question. In such obvious cases, feel free to edit it by yourself. @Justin: Nice application. Grothendieck has generalized it to schemes, when I remember correctly. To which polynomial system do you apply the result?2011-04-26
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    http://en.wikipedia.org/wiki/Ax%E2%80%93Grothendieck_theorem2011-04-26

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