Algebraic, Tropical, and Fuzzy Geometry

Andreas Dress and Walter Wenzel

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Submission date: 20. Dec. 2007
published in: Beiträge zur Algebra und Geometrie, 52 (2011) 2, p. 431-461 
DOI number (of the published article): 10.1007/s13366-011-0017-y
Bibtex
MSC-Numbers: 14A10, 08A72, 11C08, 13A18, 13F20
Keywords and phrases: Algebraic and Tropical Geometry, semiring, fuzzyring, linearly ordered abelian group, prime ideal, polynomial function, Zariski Topology, irreducible closed set

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".