On the geometry of zero sets of central quaternionic polynomials

Let R be the ring H[x₁,…,x_n] of polynomials in n central variables over the real quaternion algebra H, and let I be a left ideal in R. We prove that if p∈R vanishes at all the common zeros of I in Hⁿ with commuting coordinates, then as a slice regular quaternionic function, p vanishes at all common zeros of I in Hⁿ. This confirms a conjecture of Gori, Sarfatti and Vlacci, who settled the two dimensional case.