K-theories and Free Inductive Graded Rings in Abstract Quadratic Forms Theories

Abstract

We build on previous work on multirings ([17]) that providesgeneralizations of the available abstract quadratic forms theories (specialgroups and real semigroups) to the context of multirings ([10], [14]). Herewe raise one step in this generalization, introducing the concept of pre-specialhyperfields and expand a fundamental tool in quadratic forms theory to themore general multivalued setting: the K-theory. We introduce and developthe K-theory of hyperbolic hyperfields that generalize simultaneously Milnor’sK-theory ([11]) and Special Groups K-theory, developed by Dickmann-Miraglia ([5]). We develop some properties of this generalized K-theory, thatcan be seen as a free inductive graded ring, a concept introduced in [2] inorder to provide a solution of Marshall’s Signature Conjecture.

Keywords

• quadratic forms
• special groups
• k-theory
• multirings
• hyperfields