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 $H[q_1,\dots ,q_n]$ 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 $H[q_1,\dots ,q_n]$. 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 $H^n$.