  In this paper we prove a strong version of the Hilbert Nullstellensatz in the
ring $\mathbb H[q_1,\ldots,q_n]$ 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 $\mathbb
H[q_1,\ldots,q_n]$. 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 $\mathbb H^n$.
