# A Diophantine Decision Problem

Let $S$ be a subset of $\mathbb{R}^n$ defined by a system $\theta$ of polynomial inequalities with integer coefficients.

Let's say that $\theta$ has no integer solutions "for trivial reasons" if there is a polynomial $h$ with integer coefficients such that for all $p\in S$, it holds that $0 <h(p) < 1$.

Question: Is it effectively decidable whether a given system $\theta$ has no integer solutions "for trivial reasons"?

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Is there anyway to show that $\neg$( a given system $\theta$ has no integer solution " for trivial reason") except for pointing out a solution? –  abcdxyz Apr 12 '10 at 16:53
This system does not like the "<" sign in math. Surround such formulae by backticks, see hints in the tab on the right –  Sergei Ivanov Apr 12 '10 at 16:53
@anon: I fixed your formula, please do not remove backticks. –  Sergei Ivanov Apr 12 '10 at 16:58