Arithmetics and combinatorics of tropical Severi varieties of univariate   polynomials
  We give a description of the tropical variety of univariate polynomials of degree n having two double roots. As a set, it is given as the union of three types of maximal cones of dimension n-1, where only cones of two of these types are cones of the secondary fan of {0,...,n}. Through Kapranov's theorem, this goal is achieved by a careful study of the possible valuations of the elementary symmetric functions of the roots of a polynomial with two double root. Despite its apparent simplicity, the computation of the tropical Severi variety has both combinatorial and arithmetic ingredients. 
Alicia Dickenstein, Maria Isabel Herrero, and Luis Felipe Tabera
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