Projections of tropical varieties and an application to small tropical bases [Elektronische Ressource] / von Kerstin Hept

Projections of Tropical Varietiesand an Application toSmall Tropical BasesDissertationzur Erlangung des Doktorgradesder Naturwissenschaftenvorgelegt beim FachbereichInformatik und Mathematikder Johann Wolfgang Goethe-Universit¨atin Frankfurt am MainvonKerstin Heptaus Frankfurt am MainFrankfurt 2009(D 30)Vom Fachbereich 12 derJohann Wolfgang Goethe-Universita¨t als Dissertation angenommen.Dekan: Prof. Dr. -Ing. Detlef Kro¨mkerGutachter: Prof. Dr. rer. nat. Thorsten Theobald,Prof. Dr. sc. math. Robert BieriDatum der Disputation: 21.12.2009AbstractThe main topic of this thesis is the description of projectionsof tropical varieties and the construction of tropical bases bymeans of those projections. We present a tropical version oftheEisenbud-Evanstheorem,useso-calledregularprojectionsandcombinethemwithelimination theory. Asanapplicationof mixed fiber polytopes we obtain a description of the imageof a tropical variety. For tropical curves we deduce somebounds on the complexity of their images.Tropical geometry is a relatively new area which has its origin in the earlyseventies in the work of Bergman [Ber71] and in the middle eighties in thework of Bieri and Groves [BG84]. Given an extension field K of a valuatedfieldk BieriandGroves defineforafiniteset{a ,...,a }⊂K theBieri-Groves1 nv nset Δ (a ,...,a ) as the set of all vectors (w(a ),...,w(a )) ∈ R where w1 n 1 nKruns through all valuations of K extending the valuation v of k.