Let $N=\{1,2,\ldots,n\}$. Suppose we are given $n$ equations, with each equation taking the form $\sum\_{A\subseteq N}\left(c\_A \prod\_{i\in A}x\_i \right) = 0$, where each $c\_A$ is a real number constant. (So each equation contains at most $2^n$ terms.) An example for $n=3$ is:


$$2x\_1x\_2x\_3 - 4x\_1x\_2+5x\_3+2=0$$


$$7x\_1x\_3 - 6x\_2-4=0$$


$$-x\_1x\_2x\_3 + x\_1 - 2x\_2 +9 = 0$$


We want to find a solution $(x\_i)\_{i\in A}$ such that $0\leq x\_i\leq 1$ for all $i$ (assuming we know such a solution exists). 


Is there an algorithm that solves this in time bounded in $n$?



Solving multilinear equations
Multilinear equations are hardly easier than general equations. For instance, the multilinear equations 
 $$
\begin{cases}
x\_0-x\_1=0,\\
x\_0x\_1-x\_2=0,\\
x\_0x\_2-x\_3=0,\\
\ldots\\
x\_0x\_{n-1}-x\_n=0
\end{cases}
 $$
simply tell you that $x\_k=x\_0^k$ for all $k=1,\ldots,n$. Using this, it is very easy to replace any system of polynomial equations by a system of multilinear ones, so I assume that the most standard method (Gröbner bases) would be the main tool to use. 


