  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.

Algebraic Geometry