The perfect Nullstellensatz just got more perfect

After giving a talk about the perfect Nullstellensatz (the commutative free Nullstellensatz) at the Technion Math department’s pizza and beer seminar, I had a revelation: I think it holds over other fields as well, not just over the complex numbers! (And in particular, contrary to what I thought before, it holds over the reals. It seems to hold over other fields as well).

To explain, I will need some notation.

Let $k$ be a field. We write $A = k[z_, \ldots, z_d]$ – the algebra of all polynomials in $d$ (commuting) variables over the field $k$