Arithmetics and combinatorics of tropical Severi varieties of univariate polynomials

Abstract

Abstract. 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.

The Newton Polytope of the Morse Discriminant of a Univariate Polynomial

“…Moreover, under this change of variables, both sides of the equation ( 8) are multiplied by (M!) 2 . The latter means that the formulas for the contributions of singular points at infinity, which we are going to obtain in this subsection, will work independently of Assumption 4.13.…”

“…The degree of the Morse discriminant for general degree d univariate polynomials was computed in [10]. The tropical fan of the variety of univariate degree d polynomials having two multiple roots was studied in [2] and in [5] (in a more general setting). The maximal cones of the tropical fan, that were computed in these works, under the projection along a line spanned by a constant monomial, define the directions of all the edges of the Morse polytope.…”

“…The maximal cones of the tropical fan, that were computed in these works, under the projection along a line spanned by a constant monomial, define the directions of all the edges of the Morse polytope. However, due to non-trivial intersections of the images of the cones under this projection, the results obtained in [2] and in [5] cannot be used to enumerate the edges and vertices of the Morse polytope.…”

Characteristic classes of affine varieties and Plücker formulas for affine morphisms

“…A recent paper [DHT16] suggests a different approach to the computation of the fundamental class of the stratum 2A 1 for n = 1 (i.e. the set of all univariate polynomials with two multiple roots), and greatly clarifies the answer in this case.…”