Solving Chisini's functional equation

TL;DR

This paper analyzes the solutions of Chisini's functional equation for real functions, establishing conditions for existence, uniqueness, and continuity, and providing constructive methods for solutions under various assumptions.

Contribution

It offers necessary and sufficient conditions for solutions of Chisini's equation, including constructive methods for nondecreasing, idempotent, and continuous solutions.

Findings

Existence and uniqueness conditions for solutions of Chisini's equation.

Constructive methods for nondecreasing and continuous solutions.

Application examples demonstrating the theoretical results.

Abstract

We investigate the n-variable real functions G that are solutions of the Chisini functional equation F(x)=F(G(x),...,G(x)), where F is a given function of n real variables. We provide necessary and sufficient conditions on F for the existence and uniqueness of solutions. When F is nondecreasing in each variable, we show in a constructive way that if a solution exists then a nondecreasing and idempotent solution always exists. We also provide necessary and sufficient conditions on F for the existence of continuous solutions and we show how to construct such a solution. We finally discuss a few applications of these results.