Transcendental entire solutions of several general quadratic type PDEs and PDDEs in $ \mathbb{C}^2 $

TL;DR

This paper investigates the existence and form of transcendental entire solutions to general quadratic PDEs and PDDEs in complex two-dimensional space, extending previous results using Nevanlinna theory.

Contribution

It introduces new methods to analyze solutions of quadratic PDEs and PDDEs in several complex variables, generalizing earlier specific cases.

Findings

Characterization of solutions to quadratic PDEs in a7^2

Extension of results to more general coefficients

Application of Nevanlinna theory in multiple complex variables

Abstract

The functional equations $ f^2+g^2=1 $ and $ f^2+2\alpha fg+g^2=1 $ are respectively called Fermat-type binomial and trinomial equations. It is of interest to know about the existence and form of the solutions of general quadratic functional equations. Utilizing Nevanlinna's theory for several complex variables, in this paper, we study the existence and form of the solutions to the general quadratic partial differential or partial differential-difference equations of the form $ af^2+2\alpha fg+b g^2+2\beta f+2\gamma g+C=0 $ in $ \mathbb{C}^2 $. Consequently, we obtain certain corollaries of the main results of this paper concerning binomial equations which generalize many results in [\textit{Rocky Mountain J. Math.} \textbf{51}(6) (2021), 2217-2235] in the sense of arbitrary coefficients.