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Existence of Solutions to Systems of General Quadratic Functional Equations in $\mathbb{C}^n$

Molla Basir Ahamed, Sanju Mandal

TL;DR

The paper studies the existence and form of finite-order transcendental solutions to systems of general quadratic functional equations in ${\mathbb C}^n$ using Nevanlinna theory in several complex variables and its difference analogues. It analyzes three equation types—difference, partial differential, and partial differential-difference—under a nondegeneracy condition ${\Delta} \neq 0$ and derives explicit cosine–sine solution forms when possible, while proving nonexistence for a pure PDE system and providing explicit solutions for the difference and PDE–difference systems with precise parameter constraints. Corollaries extend the results to arbitrary coefficients and higher dimensions ($n\ge 2$), and illustrative examples confirm the existence of transcendental finite-order solutions. Overall, the work advances the understanding of how general quadratic structures constrain meromorphic solutions in several complex variables and broadens the scope beyond earlier work on fermat-type and trinomial equations.

Abstract

The main objective of this study is to investigate the existence and forms of solutions of systems of 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 and partial differential-difference equations of the form $af^2 + 2αfg + bg^2 + 2βf + 2γg + C=0$, where $f$ and $g$ are non-constant meromorphic functions in $\mathbb{C}^n$. The obtained results in this article are improvements and generalizations of several results from [\textit{RACSAM}, \textbf{116}(8) (2022)]. Furthermore, appropriate remarks and illustrative examples are provided to validate and demonstrate the applicability of the obtained results concerning the existence and forms of solutions for such systems of equations.

Existence of Solutions to Systems of General Quadratic Functional Equations in $\mathbb{C}^n$

TL;DR

The paper studies the existence and form of finite-order transcendental solutions to systems of general quadratic functional equations in using Nevanlinna theory in several complex variables and its difference analogues. It analyzes three equation types—difference, partial differential, and partial differential-difference—under a nondegeneracy condition and derives explicit cosine–sine solution forms when possible, while proving nonexistence for a pure PDE system and providing explicit solutions for the difference and PDE–difference systems with precise parameter constraints. Corollaries extend the results to arbitrary coefficients and higher dimensions (), and illustrative examples confirm the existence of transcendental finite-order solutions. Overall, the work advances the understanding of how general quadratic structures constrain meromorphic solutions in several complex variables and broadens the scope beyond earlier work on fermat-type and trinomial equations.

Abstract

The main objective of this study is to investigate the existence and forms of solutions of systems of general quadratic functional equations in . By utilizing Nevanlinna theory in , we explore the existence and form of solutions for the several systems of general quadratic difference and partial differential-difference equations of the form , where and are non-constant meromorphic functions in . The obtained results in this article are improvements and generalizations of several results from [\textit{RACSAM}, \textbf{116}(8) (2022)]. Furthermore, appropriate remarks and illustrative examples are provided to validate and demonstrate the applicability of the obtained results concerning the existence and forms of solutions for such systems of equations.
Paper Structure (10 sections, 5 theorems, 173 equations)

This paper contains 10 sections, 5 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Background of Circular-type equations and their solutions in several complex variables:
  3. Background of quadratic trinomial equations and their solutions in one and several complex variables:
  4. System of quadratic equations and their solutions in several complex variables:
  5. Observations:
  6. Some notations and Key lemmas
  7. Main results and its proof
  8. System of GQE type difference equations:
  9. System of GQE type partial differential equations:
  10. System of GQE type partial differential-difference equations:

Key Result

Theorem 3.1

Let $c=(c_1,\ldots,c_n)\in\mathbb{C}^n \setminus\{(0,\ldots,0)\}$ and $a, b, C, \alpha, \beta, \gamma\in\mathbb{C}$ such that $ab\neq 0$, $\Delta\neq 0$, and $\alpha^2\neq 0, ab$. Then the pair of finite order transcendental entire solutions in $\mathbb{C}^n$ for the system of general quadratic diff must be one of the forms where $\gamma(z)=L(z)+\Psi(z)$, $L(z)$ is a linear function of the form $

Theorems & Definitions (14)

  • Theorem 3.1
  • proof : Proof of Theorem \ref{['th-2.1']}
  • Remark 3.1
  • Example 3.1
  • Remark 3.2
  • Corollary 3.1
  • Theorem 3.2
  • proof : Proof of Theorem \ref{['th-2.2']}
  • Remark 3.3
  • Theorem 3.3
  • ...and 4 more