Algebraic, tropical, and fuzzy geometry

Abstract

The relatively new field of "Tropical Geometry" and "Classical Algebraic Geometry" have much in common. In this paper, we present a unified approach to both of these fields by studying more general "Fuzzy Geometries" over fuzzy rings. It turns out that the fuzzy ring which underlies "Tropical Geometry" is the same as that belonging to "Valuated Matroids".

Arithmetic and Polynomials over Fuzzy Rings

“…In this section, we first make available the basic definitions concerning fuzzy rings as stated in [2] and [4].…”

“…In [4], we have already studied the Zariski topology for more general classes of functions and not only for those induced by polynomials. Note thatas even in classical ring theory and in classical field theory-one has to distinguish between polynomials and polynomial functions.…”

“…Roughly speaking, Tropical Geometry is the geometry over the semiring of real numbers with the operations of tropical addition, which means taking the minimum of two numbers, and of tropical multiplication, that is ordinary addition. In subsequent papers-as in [4], [6], [7]-the rôle of the minimum is often replaced by the maximum, just by inverting the usual ordering defined on . Sometimes this is more convenient by studying relations to Non-Archimedian Valuations.…”

“…for instance [5]) and "Tropical Geometry" on the other handside have much in common. Therefore, we have already established a unified approach to both of these fields in [4]. The appropriate objects we have studied are so-called "Fuzzy Geometries" over "Fuzzy Rings."…”

“…In [4], we have already studied "Fuzzy Geometries" over quasi fuzzy domains as well as more general Zariski Topologies. Contrary to the present paper, these Zariski Topologies are defined in terms of sets of functions; these are not necessarily polynomial functions.…”