(Joint work with Kaique M.A. Roberto (IME-USP) and Hugo R.O. Ribeiro (IME-USP))
A superring is essentially a ring with multivalued sum and product: for instance, the set of polynomials with coefficients in a multiring has a natural structure of superring.
In the first part of the talk, we consider some constructions of superrings, explore some properties of the superring of polynomials with coefficients in a hyperfield and develop some fragments of the theory of algebraic extension for superfields, such as the addition of roots to a superfield.
The significance of these multialgebraic methods to (univalent) Commutative Algebra is indicated by applying these results to algebraic theory of quadratic forms:
(i) obtaining new relevant constructions in the category of special groups (or its equivalent category special hyperfields);
(ii) extending the validity of the Arason-Pfister Hauptsatz - a positive answer to a question posed by Milnor in a classical paper of 1970 - and established by Dickmann-Miraglia in the realm of reduced special groups (or its equivalent category real reduced hyperfields) in 2000.
In the second part of the talk, we expand a fundamental tool in algebraic theory of quadratic forms to the more general multivalued setting: the K-theory. We introduce and develop the K-theory of hyperbolic hyperfields that generalize simultaneously Milnor's K-theory and Special Groups K-theory, developed by Dickmann-Miraglia. We develop some properties of this generalized K-theory, that can be seen as a free inductive graded ring, a concept introduced in order to provide a solution of Marshall's Signature Conjecture. Moreover, the extended version of Arason-Pfister Hauptsatz entails some interesting properties concerning K-theory and Marshall's conjecture.
We finish this series of talks indicating some future developments of superrings theory and possible applications.