In this paper we prove a strong version of the Hilbert Nullstellensatz in the ring
of slice regular polynomials in several quaternionic variables. Our proof deeply depends on a detailed analysis of the common zeros of slice regular polynomials which belong to an ideal in the ring of slice regular polynomials in several quaternionic variables. This study motivates the introduction of a new notion of algebraic set in the quaternionic setting, which allows us to define a Zariski-type topology on H^n
.
