A short survey on Newton polytopes, tropical geometry and ring of conditions of algebraic torus

TL;DR

This paper provides an accessible overview of tropical and toric geometry, focusing on combinatorial and convex geometric methods used to understand solutions of polynomial systems and the ring of conditions of algebraic tori.

Contribution

It offers a clear exposition of recent concepts in tropical and toric geometry, emphasizing their combinatorial and convex geometric aspects for a broad mathematical audience.

Findings

Descriptions of the ring of conditions of algebraic torus

Connections between algebraic geometry and convex geometry

Accessible exposition for students and mathematicians

Abstract

The purpose of this note is to give an exposition of some interesting combinatorics and convex geometry concepts that appear in algebraic geometry in relation to counting the number of solutions of a system of polynomial equations in several variables over complex numbers. The exposition is aimed for a general audience in mathematics and we hope to be accessible to undergraduate as well as advance high school students. The topics discussed belong to relatively new, and closely related branches of algebraic geometry which are usually referred to as tropical geometry and toric geometry. These areas make connections between the study of algebra and geometry of polynomials and the combinatorial and convex geometric study of piecewise linear functions. The main results discussed in this note are descriptions of the so-called "ring of conditions" of algebraic torus.