Tropical Resultants for Curves and Stable Intersection

TL;DR

This paper introduces tropical resultants for curves, demonstrating how they can be used to compute stable intersections in tropical geometry, linking algebraic and tropical intersections under generic conditions.

Contribution

It defines tropical resultants for planar curves and establishes their use in computing stable intersections, connecting algebraic and tropical geometry.

Findings

Tropical resultants can compute stable intersections of tropical curves.

Stable intersection corresponds to algebraic intersection under generic conditions.

Conditions on residual coefficients ensure the validity of the tropicalization approach.

Abstract

We introduce the notion of resultant of two planar curves in the tropical geometry framework. We prove that the tropicalization of the algebraic resultant can be used to compute the stable intersection of two tropical plane curves. It is shown that, for two generic preimages of the curves to an algebraic framework, their intersection projects exactly onto the stable intersection of the curves. It is also given sufficient conditions for such a generality in terms of the residual coefficients of the algebraic coefficients of defining equations of the curves.