# Non Commutative Algebraic Geometry I: Monomial Equations with a Single Variable

## TL;DR

This paper introduces a framework for analyzing solutions to monomial equations with a single variable over free associative algebras, using algebraic geometry tools like Makanin-Razborov diagrams.

## Contribution

It develops the first part of a series on solution structures, constructing diagrams that encode all homogeneous solutions to such systems.

## Key findings

- Constructed Makanin-Razborov diagrams for homogeneous solutions
- Analyzed solution sets of monomial equations with one variable
- Laid groundwork for further algebraic geometry studies

## Abstract

This paper is the first in a sequence on the structure of sets of solutions to systems of equations over a free associative algebra. We start by constructing a Makanin-Razborov diagram that encodes all the homogeneous solutions to a homogeneous system of equations. Then we analyze the set of solutions to monomial systems of equations with a single variable.