Problem: I have a set of $m$ multivariate polynomials over $k$ variables, with an upper bound on the degrees $n$:
$$ \{f_i(x_1,x_2...x_k)\}_{i=1}^{m} $$
My goal is to find if there's a real root such that for all $j$, $0 \leq x_j \leq 1$ i.e. the root is in the k-dimensional hypercube.
What I know: I know that this can be symbolically solved using Gröbner bases but the operation is too costly for my purposes since I need to do this multiple times. Right now I'm doing this numerically with the Newton-Raphson method but sometimes it converges to a root outside the hypercube even if one exists inside and I don't have an intuition for a heuristic starting point picking algorithm. I was able to find a multidimensional generalization of the Bisection method but I don't have the guarantee that the function will have differing signs at the vertices of the hypercube. I was wondering if there's something like a multidimensional/multivariate version of Sturm's theorem so I can solve the problem without explicitly finding the roots. This question indicates that it is not possible but it is 13 years old and I don't have the intuition to evaluate the answers provided.
Thanks for your help.