I'm looking to find a root $(x_1,\dots,x_n)$ of a polynomial $p \in {\mathbb R}[x_1,\dots,x_n]$ such that $0 \leq x_i < 1$ for all $i$. Further, I know in advance that setting $x_1 = \cdots = x_n$ is a root of $p$, but wish to avoid this root. How can I find at least one of these roots, preferably using a computer?

In more detail, $p$ is formed by taking two homogeneous polynomials $q_1, q_2$ (of degrees $d_1$ and $d_2$) where every coefficient is either zero or one, letting $D$ be the least common multiple of $d_1$ and $d_2$, and setting $p = q_1^{D/d_1} - q_2^{D/d_2}$. This is encoding the simultaneous solution to the equations $$q_1(x) = y^{d_1}, \quad q_2(x) = y^{d_2}$$ over $(x_1,\dots,x_n, y) \in [0,1)^{n+1}$.