Number of common roots and resultant of two tropical univariate polynomials

TL;DR

This paper extends the classical relationship between common roots and resultants from complex polynomials to tropical polynomials, showing the equivalence under certain conditions.

Contribution

It demonstrates that the fundamental link between roots and resultants persists in tropical polynomial theory with appropriate definitions.

Findings

The number of common roots equals the order of the tropical resultant for simple, non-zero roots.

The classical root-resultant relationship is valid in the tropical setting under specific conditions.

Abstract

It is well known that for two univariate polynomials over complex number field the number of their common roots is equal to the order of their resultant. In this paper, we show that this fundamental relationship still holds for the tropical polynomials under suitable adaptation of the notion of order, if the roots are simple and non-zero.