Conditions for solving polynomial equations using algebraic and 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.