Tropical Effective Primary and Dual Nullstellens\"atze

TL;DR

This paper proves a tropical Nullstellensatz and its effective version, advancing the algebraic and computational understanding of tropical polynomials and their applications in mathematics.

Contribution

It introduces a tropical Nullstellensatz and an effective formulation, along with simple dualities and connections to min-plus systems, enhancing the algebraic theory of tropical polynomials.

Findings

Proved a tropical Nullstellensatz.

Established an effective version of the Nullstellensatz.

Connected tropical and min-plus polynomial systems.

Abstract

Tropical algebra is an emerging field with a number of applications in various areas of mathematics. In many of these applications appeal to tropical polynomials allows to study properties of mathematical objects such as algebraic varieties and algebraic curves from the computational point of view. This makes it important to study both mathematical and computational aspects of tropical polynomials. In this paper we prove a tropical Nullstellensatz and moreover we show an effective formulation of this theorem. Nullstellensatz is a natural step in building algebraic theory of tropical polynomials and its effective version is relevant for computational aspects of this field. On our way we establish a simple formulation of min-plus and tropical linear dualities. We also observe a close connection between tropical and min-plus polynomial systems.