Arithmetics and combinatorics of tropical Severi varieties of univariate polynomials

TL;DR

This paper characterizes the tropical variety of univariate degree n polynomials with two double roots, revealing its structure as a union of specific cones and exploring the combinatorial and arithmetic aspects involved.

Contribution

It provides a detailed description of the tropical Severi variety for polynomials with two double roots, including its geometric structure and valuation analysis.

Findings

The tropical Severi variety is a union of three types of maximal cones.

Only two types of these cones are part of the secondary fan.

The study combines combinatorial and arithmetic methods to analyze valuations.

Abstract

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.