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

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