Say that we have an $n$ dimensional polynomial of degree $m$. Are there any methods to check whether a root exists when $x_k \in [a_k, b_k]$ other than directly attempting Newton's method to solve for one?
Again I am only trying to prove the existence of at least one root in the set of intervals.
I'm going to link this to a similar, recent question below which is not exactly a duplicate but leads, in my opinion, to the same answer: to use the Interval Arithmetic methods,
Intervals and polynomials