Conditions for solving polynomial equations using algebraic and hypergeometric functions

TL;DR

This paper explores the conditions under which polynomial equations of degree greater than six can be solved using algebraic and hypergeometric functions, providing proofs on solvability and non-solvability based on degree and coefficients.

Contribution

It offers new proofs regarding the non-existence of algebraic solutions for degrees greater than four and establishes conditions for solving higher-degree equations with hypergeometric functions.

Findings

Algebraic solutions do not exist for degree >4 equations.

Equations of degree >5 require specific coefficient conditions for hypergeometric solutions.

General trinomial equations can be solved using hypergeometric functions.

Abstract

In this paper, we focus on clarifying the concept of solving equations of degree greater than six using continuous functions or hypergeometric functions and providing another proof of the non-existence of algebraic solutions for equations of degree greater than four. According to the Kolmogorov-Arnold theorem, we will prove that equations of degree greater than five cannot be solved without special conditions between their coefficients using hypergeometric functions. However, we prove that trinomial equations of general form can in general be solved using hypergeometric functions.