This paper presents a new algorithm for solving a system of polynomials, in a domain of $ ^n$. It can be seen as an improvement of the Interval Projected Polyhedron algorithm proposed by Sherbrooke and Patrikalakis . It uses a powerful reduction strategy based on univariate root finder using Bernstein basis representation and Descarte's rule. We analyse the behavior of the method, from a theoretical point of view, shows that for simple roots, it has a local quadratic convergence speed and gives new bounds for the complexity of approximating real roots in a box of $ ^n$. The improvement of our approach, compared with classical subdivision methods, is illustrated on geometric modeling applications such as computing intersection points of implicit curves, self-intersection points of rational curves, and on the classical parallel robot benchmark problem.