Characterization of solutions of refined Fermat-type functional equations in $ \mathbb{C}^n $

TL;DR

This paper investigates the existence and explicit forms of transcendental solutions to refined Fermat-type functional equations with polynomial coefficients in several complex variables, using Nevanlinna theory.

Contribution

It provides new results on the solutions of quadratic trinomial and binomial equations in $\\mathbb{C}^n$, extending previous studies and exploring conditions for non-existence.

Findings

Solutions of certain quadratic trinomial equations in $\mathbb{C}^n$ are characterized.

Existence conditions for solutions of binomial equations are established.

Examples support the theoretical results and demonstrate cases of non-existence.

Abstract

The main purpose of this article is concerned with the existence and the precise forms of the transcendental solutions of several refined versions of Fermat-type functional equations with polynomial coefficients in several complex variables by utilizing the Nevanlinna theory of meromorphic functions in several complex variables. In fact, we investigate the existence and forms of the transcendental solutions of non-linear quadratic trinomial equations in $\mathbb{C}^n$. As a consequence of our result, we show that solutions of binomial equations in $\mathbb{C}^n$ can be explored and this exploration broaden the scope of the study of functional equations in $ \mathbb{C}^n $. The results we obtained are improvements over certain recent findings, noted as remarks. In addition, some examples relevant to the content of the paper have been exhibited in support of the validation of each results. Further, we discuss situations when solutions of such equations does not exist.