I will discuss two versions of the Nullstellensatz in the setting of nc function theory. The first one is a perfect free commutative Nullstellensatz, which we recently (re)-discovered apropos our work on the classification of functions of bounded nc analytic functions. This Nullstellensatz allows to classify quotients of the algebra of commutative polynomials in a crystal clear and intuitive fashion, in terms of noncommutative varieties.

The second Nullstellensatz is a new variant of an analytic Nullstellensatz for homogeneous varieties, which has a classical flavour, and was proved by Davidson, Ramsey and I when we worked on the isomorphism problem of complete Pick algebras. I will explain how this Nullstellensatz is used to classify algebras of bounded analytic nc functions on commutative nc varieties.

This talk is based on joint works with Guy Salomon and Eli Shamovich.