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Hugo Rafael de Oliveira Ribeiro

Publications and source records attributed to Hugo Rafael de Oliveira Ribeiro.

5 recordsLinked to original sources
arXiv2024

The general Arason-Pfister Hauptsatz

In the present we develop a fragment of the theory of superfields, polynomials and Marshall's quotient in order to obtain for general special groups, a proof of the Arason-Pfister Hauptsatz (APH): "if $ϕ\neq \emptyset$ is an anisotropic form and $ϕ\in I^n(F)$ then $dim (ϕ) \geq 2^n$". In the process, we also obtain an alternative proof of APH for reduced special groups that avoid the uses of the invariants developed in \cite{dickmann2000special}. The applications of the full Arason-Pfister Hauptsatz leads to interesting properties of graded rings associated to special groups/hyperfields. \textbf{Keywords:} Arason-Pfister Hauptsatz; hyperfields; special groups; Milnor K-theory; graded rings.

math.AC↗
arXiv2023

Linear Systems, Matrices and Vector Spaces over Superfields

Motivated by some recent developments in abstract theories of quadratic forms, we start to develop in this work an expansion of Linear Algebra to multivalued structures (a multialgebraic structure is essentially an algebraic structure but endowed with some multivalued operations). We introduce and study matrices and determinants over a commutative superrings (roughly, a ring where the sum and product are multivalued) and study linear systems and vector spaces over superfields. As an application, we obtain a fundamental result to the development of a theory of algebraic extensions of superfields.

math.AC↗
arXiv2023

On algebraic extensions and algebraic closures of superfields

Building over recent results, we expand the basic theory of algebraic extensions to the realm of superfields -a field with multivalued sum and product-, showing that every superfield has a (unique up to isomorphism) strong algebraic extension to a superfield that is algebraically closed. Moreover we show that every infinite algebraically closed superfield admits quantifier elimination procedure.

math.AC↗
arXiv2021

von Neumann regular Hyperrings and applications to Real Reduced Multirings

A multiring ([Mar3]) is a kind of ring where is allowed the sum of two elements to be anon-empty subset of the structure instead of just one element -and an hyperring is a multiring with a strong distributive property. Thus a reduced hyperring where the prime spec is a Boolean topological space is called von Neumann regular hyperring (vNH). It is possible to associate to every such object a structural presheaf in the same way it is made with rings but there are some vNH such that this presheaf is not a sheaf. In this sense, we give a first-order characterization of vNH with a structural sheaf (geometric vNH or just GvNH) and how to transform a vNH in aGvNH -in fact, this transformation shows that the category GvNH is a reflexive subcategory of vNH. We also build a von Neumann regular hull for multirings and use this to give applications for algebraic theory of quadratic forms. More precisely, we work with Real Reduced Multiring (RRM, [Mar3]) -also known as Real Semigroup (RS, [DP1])-, a special kind of multirings that is useful to explore the real structure of rings, and show that a von Neumann hull of a RRM is again a RRM. This gives a generalization of sheafs arguments present in [DM4].

math.AC↗