Tropical algebraic geometry is the geometry of the tropical semiring (R, min, +). Its objects are polyhedral cell complexes which behave like complex algebraic varieties. We give an introduction to this theory, with an emphasis on plane curves and linear spaces. New results include a complete description of the families of quadrics through four points in the tropical projective plane and a counterexample to the incidence version of Pappus' Theorem.
The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. This research monograph has three goals. First of all it serves as a comprehensive source for all results that I have been able to obtain in connection to the Universality Theorem for 4-polytopes. It includes complete proofs of all these results including a proof of the Universality Theorem itself. Secondly, it is (as the title says) meant as an introduction to the beautiful theory of realization spaces of polytopes. For that purpose also a treatment of Steinitz's Theorem is included. Although the result is classical the proof presented here contains some new and fresh elements. In particular, we provide a new proof for Tutte's Theorem on equilibrium representations of planar graphs. We also give a complete proof of Mnev's Universality Theorem for oriented matroids (and of its generalization: the Universal Partition Theorem). Last but not least, this monograph is written for the sake of enjoyment of geometric constructions. Most of the concepts and constructions that are needed here are elementary in nature. The final construction for the Universality Theorem is obtained by building larger and larger polytopal units of increasing geometric and algebraic complexity. We start from small incidence configurations, go to polytopes for addition and multiplication, and end up with polytopes that encode entire polynomial inequality systems. I hope that the reader can feel the fun that lies in these constructions.There are many alternative ways of approaching the main results of this monograph. In particular, there are several different ways to build up the proof viii PREFACE of the Universality Theorem for 4-polytopes. However, all the approaches kown to me rely on similar principles:• first construct small and useful polytopes (using Lawrence Extensions or similar techniques) that have non-prescribable facets (or vertex figures),• use connected sums to join these polytopes to larger units that are capable of encoding arithmetic operations,• finally use connected sums to join these arithmetic units into even larger polytopes that encode entire polynomial inequality systems.Here I have chosen an approach that is very modular. The basic building blocks are very simple polytopes, and the whole complexity is governed by the way of composing these blocks.In order to obtain the strongest possible results it was necessary to set up a new concept of stable equivalence that compares realization spaces with other semialgebraic sets. The reader may excuse the fact that whenever stable equivalence between two spaces is proved the exposition becomes a bit technical. Everywhere else I used concrete geometric approaches rather than abstract settings. Whenever it is possible the constructions are carried out in an explicit manner. There are many people who have made the wri...
The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
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