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 $n$?

Exactly, no. With the trick in Vladimir Dotsenko's answer you can easily write a degree-5 polynomial equation in this form, by adding extra variables for $x^2, x^3, x^4$. Evil Abel foils our plans again. $\endgroup$ – Federico Poloni May 28 at 16:50