A Strong Version of the Hilbert Nullstellensatz for slice regular polynomials in several quaternionic variables

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