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"?