Polynomial Equations: Theory and Practice

TL;DR

This paper explores the theory and practical methods for solving polynomial equations, focusing on algebraic structures, solution counts, and software tools to address polynomial optimization challenges.

Contribution

It introduces systems of polynomial equations, discusses main solution approaches, and illustrates the theory with diverse examples and software applications.

Findings

Analysis of critical point equations

Discussion of algebraic varieties and solution counts

Illustrative examples using various software packages

Abstract

Solving polynomial equations is a subtask of polynomial optimization. This article introduces systems of such equations and the main approaches for solving them. We discuss critical point equations, algebraic varieties, and solution counts. The theory is illustrated by many examples using different software packages.