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 be a field. We write – the algebra of all polynomials in (commuting) variables over the field . 

Topics in Operator Theory, Lecture 6: an overview of noncommutative boundary theory

The purpose of this lecture is to introduce some classical notions in uniform algebras that motivated Arveson’s two seminal papers, “Subalgebras of C*-algebras I + II”, and then to introduce the basic ideas on how to generalize to the noncommutative setting, which were introduced in those papers.

Note: If you are following the notes of this course, please note that the previous lecture has been updated with quite a lot of material. 

The perfect Nullstellensatz

Question: to what extent can we recover a polynomial from its zeros?

Our goal in this post is to give several answers to this question and its generalisations. In order to obtain elegant answers, we work over the complex field (e.g., there are many polynomials, such as , that have no real zeros; the fact that they don’t have real zeros tells us something about these polynomials, but there is no way to “recover” these polynomials from their non-existing zeros). We will write for the algebra of polynomials in one complex variable with complex coefficients, and consider a polynomial as a function of the complex variable . We will also write for the algebra of polynomials in (commuting) variables, and think of polynomials in – at least initially – as a functions of the variable . 

[Update June 24, 2019: contrary to what I thought, the main theorem presented below holds over arbitrary fields, not just over the complex numbers, very much by the same proof. See this post.]