The tropical discriminant in positive characteristic

TL;DR

This paper develops methods to analyze singularities in tropical hypersurfaces over fields of positive characteristic, providing computational tools and explicit descriptions of tropical discriminants and their geometric features.

Contribution

It introduces a novel approach to compute singular points and discriminants of tropical hypersurfaces in positive characteristic, extending tropical geometry techniques.

Findings

Computed singular points of tropical hypersurfaces in positive characteristic.

Determined all maximal cones of tropical linear spaces with fixed double roots.

Counted vertices, edges, and faces of Newton polytopes of discriminants in characteristic p.

Abstract

We study singularities in tropical hypersurfaces defined by a valuation over a field of positive characteristic. We provide a method to compute the set of singular points of a tropical hypersurface in positive characteristic and the p-adic case. This computation is applied to determine all maximal cones of the tropical linear space of univariate polynomials of degree $n$ and characteristic $p$ with a fixed double root and the fan of all tropical polynomials that have $0$ as a double root independently of the characteristic. We also compute, by pure tropical means, the number of vertices, edges and 2-faces of the Newton polytope of the discriminant of polynomials of degree $p$ in characteristic $p$.