Solutions for several systems of algebraic differential-difference equations in $\mathbb{C}^n$

Abstract: In this article, we introduce a new method for solving general quadratic functional equations in $\mathbb{C}^n$. By utilizing Nevanlinna theory in $\mathbb{C}^n$, we explore the existence and form of solutions for the several systems of general quadratic difference or partial differential-difference equations of the form $af^2 + 2\alpha fg + bg^2 + 2\beta f + 2\gamma g + C=0$, where $f$ and $g$ are non-constant meromorphic functions in $\mathbb{C}^n$. The results in this paper are some improvements and generalizations of several existing results on the existence and forms of solutions for these systems. Specifically, the existing results were related to Circular-type $ f^2+g^2=1 $ and quadratic trinomial functional equations $ f^2+2\alpha fg+g^2=1 $. We provide examples to illustrate the validity of our conclusions regarding the existence and forms of solutions for such system of equations.