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Show HONORS COLLEGE SPRING 2013 Isabelle Scott Tommaso de Fernex 242 TROPICAL ANALOG TO THE LOG CANONICALTHRESHOLD Isabelle Scott (Tommaso de Fernex) Department of Mathematics University of Utah In algebraic geometry, the log canonical threshold is a property of singularities of planar curves. While singularities have multiplicities, the log canonical threshold can be a m o r e telling invariant. It helps to classify curves beyond what the multiplicity indicates by examining h o w quickly the inverse of a singularity goes to infinity. Tropical geometry is a fairly new field where algebra is considered over the extended real numbers with the t w o binary operations of m i n i m u m and addition. This algebra forms a semiring where the geometry of curves b e c o m e s piece-wise linear. M a n y equivalences to different concepts in algebraic geometry have been considered tropically, as it is often a w a y to simplify calculations. In this paper we first explain both of these concepts in more depth and then propose a tropical equivalence to the log canonical threshold and examine the extent to which this analog holds. W e consider tropical polynomials and use the dual polygon of the graphs determined by these polynomials to study our definition of tropical log canonical threshold. W e then prove s o me properties of this n e w concept that help to relate it to the algebraic case. W e examine curves that have clear algebraic parallels, such as the intersection of three lines, a nd compare the log canonical thresholds in each setting. |
