# Amoebas and Laurent series

Abstract: The amoeba of a polynomialpis the logarithmic image of the zero set of thepolynomial. l/pcan be developed into several Laurent series. Thelogarithmic image of every such Laurent series have a domain ofconvergence and its loga rithmic image is a component of thecomplement of the amoeba of p. This is the background for thethesis subjett. The Thesis begins with an introduction toLaurent series in general and contains, for instance, a proofof the logarithmic convexity of the domain of convergence. Thenin Chapter 2 we present general fatts about arnoebas. Theconnection between components of the axnoeba com plement andinteger points of the Newton polytope is obtained. In Chapter 3we give a relatively complete account of amoebas of polynomialswhich is a product of afline polynomials. The zero set of sucha polynomial is an arrangement of hyperplanes. The amoebas isthen a union of the amoebas of individual hyper planes. Wedescribe these individual amoebas and their complements. Wefind that if the arrangement of amoebas is in general positionthe number of compo nents is the same as the number of integerpoints in the Newton polytope and the homology of thecomplement of the hyperplane arrangement is generated by thecycles which comes from the different components of the amoebacomplement. In Chapter 4 we consider fundamental periods andtheir connection with Laurent series. It turns out that afundamental period is a coefficient of a Laurent series. Themain result here is that some fundamental periods which areintegrals over cycles of dimension n can be reduced to certeinlower dimensional integrals. The last chapter of the first partpresents amoebas from a heuristic viewpoint using computergenerated pictures of some amoebas. The main idea is that theamoebas seem to contain a dual graph of some triangulation ofthe Newton polytope. These dual graphs we tall crackingpatterns and we give some idas of this object in the rest ofCapter 5. The setond part of the thesis is the paperArrangements of hyperplaneamoebas which is writtenjointly with Mikael Passare and August Tsikh. This paper dealswith the compactified situation which parallels the situationin Chapter 3.

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