Tropical analog to the log canonical threshold

honors thesisCollege of ScienceMathematicsTommaso de FernexIn 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 more telling invariant. It helps to classify curves beyond what the multiplicity indicates by examining how 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 two binary operations of minimum and addition. This algebra forms a semi-ring where the geometry of curves becomes piece-wise linear. Many equivalences to different concepts in algebraic geometry have been considered tropically, as it is often a way 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. We consider tropical polynomials and use the dual polygon of the graphs determined by these polynomials to study our definition of tropical log canonical threshold. We then prove some properties of this new concept that help to relate it to the algebraic case. We examine curves that have clear algebraic parallels, such as the intersection of three lines, and compare the log canonical thresholds in each setting