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Abstract <p>In this paper, we mainly investigate the existence and forms of entire solutions to Fermat-type trinomial partial differential-difference equations in <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7244_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="18" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb{C}^{2}\)</EquationSource> <!--ContMath2460210Zhang-m3--> </InlineEquation>. Utilizing Nevanlinna’s theory of meromorphic functions in several complex variables, we obtain the general forms of entire solutions for such equations, which extend and generalize some previous results from Fermat-type equations in one or several complex variables. Additionally, we provide some examples to illustrate our conclusions.</p>

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On Entire Solutions of Fermat-Type Partial Differential-Difference Equations in \(\boldsymbol{\mathbb{C}^{2}}\)

  • 期刊论文

  • 发表日期：

    2025年10月10日

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  • J. Zhang,
  • Z. Huang

摘要

Abstract

In this paper, we mainly investigate the existence and forms of entire solutions to Fermat-type trinomial partial differential-difference equations in \(\mathbb{C}^{2}\) . Utilizing Nevanlinna’s theory of meromorphic functions in several complex variables, we obtain the general forms of entire solutions for such equations, which extend and generalize some previous results from Fermat-type equations in one or several complex variables. Additionally, we provide some examples to illustrate our conclusions.