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It is clear that the following problem is in $\mathsf{NP}_\mathbb{R}$.

Input: a list $P$ of triplets $(a,s,t)$ where $s$ and $t$ are nonnegative integers.
Output: is there an $x\in \mathbb{R}$ such that $$p(x) =\sum_{(a,s,t) \in P} a x^s(1-x)^t \geq 0?$$

Is this problem $\mathsf{NP}_\mathbb{R}$-hard?

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    $\begingroup$ The question is essentially if the given $-p(x)$ is not always positive so Positivstellensatz might be related. $\endgroup$ – Kaveh Jun 30 '15 at 6:17

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