Algebraic Geometry Over Hyperrings

By Jaiung Jun. 2015

74 citations recorded by the source corpus.

We develop basic notions and methods of algebraic geometry over the algebraic objects called hyperrings. Roughly speaking, hyperrings generalize rings in such a way that an addition is `multi-valued'. This paper largely consisits of two parts; algebraic aspects and geometric aspects of hyperrings. We first investigate several technical algebraic properties of a hyperring. In the second part, we begin by giving another interpretation of a tropical variety as an algebraic set over the hyperfield which canonically arises from a totally ordered semifield. Then we define a notion of an integral hyperring scheme $(X,\mathcal{O}_X)$ and prove that $\Gamma(X,\mathcal{O}_X)\simeq R$ for any integral affine hyperring scheme $X=Spec R$.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ac8ca44da897971fba324a03c42eca0eac6ae539

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Matroids over hyperfields (2016, background): “Although matroids over fuzzy rings are somewhat more general than matroids over hyperfields, we believe our work has some advantages over the Dress–Wenzel theory, including the fact that (according to MathSciNet) few authors besides Dress and Wenzel themselves have studied or used their notion of fu”