Arithmetics and combinatorics of tropical Severi varieties of univariate polynomials

LUIS TABERA; LUIS TABERA; ALICIA DICKENSTEIN; MARÍA ISABEL HERRERO; ALICIA DICKENSTEIN; MARÍA ISABEL HERRERO

ISRAEL JOURNAL OF MATHEMATICS

HEBREW UNIV MAGNES PRESS

Lugar: Jerusalem; Año: 2017 vol. 221 p. 741 - 777

0021-2172

We give a description of the tropical Severi variety of univariate polynomials of degree n having two double roots. We show that, as a set, it is given as the union of three explicit types of cones of maximal 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 roots. Despite its apparent simplicity, the computation of the tropical Severi variety has both combinatorial and arithmetic ingredients.