Solutions of Complex Fermat-Type Partial Difference and Differential-Difference Equations

Abstract

The functional equation \(f^{m}+g^{m}=1\) can be regarded as the Fermat-type equations over function fields. In this paper, we investigate the entire and meromorphic solutions of the Fermat-type functional equations such as partial differential-difference equation \(\left( \frac{\partial f(z_{1}, z_{2})}{\partial z_{1}}\right) ^{n}+f^{m}(z_{1}+c_{1}, z_{2}+c_{2})=1\) in \(\mathbb {C}^{2}\) and partial difference equation \(f^{m}(z_{1}, \ldots , z_{n})+f^{m}(z_{1}+c_{1}, \ldots , z_{n}+c_{n})=1\) in \(\mathbb {C}^{n}\) by making use of Nevanlinna theory for meromorphic functions in several complex variables.

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  • 04 December 2019

    We give a correction to Theorem 1.2 in a previous paper [Mediterr. J. Math. (2018) 15:227]. Two examples are given to explain the corrected conclusion.

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Correspondence to Tingbin Cao.

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This research was supported by the National Natural Science Foundation of China (No.11871260, No.11461042), the Natural Science Foundation of Jiangxi Province (No.20161BAB201007) and the outstanding young talent assistance program of Jiangxi Province (No.20171BCB23002) in China.

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Xu, L., Cao, T. Solutions of Complex Fermat-Type Partial Difference and Differential-Difference Equations. Mediterr. J. Math. 15, 227 (2018). https://doi.org/10.1007/s00009-018-1274-x

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Keywords

  • Several complex variables
  • meromorphic functions
  • Fermat-type equations
  • Nevanlinna theory
  • partial differential-difference equations

Mathematics Subject Classification

  • Primary 39A45
  • Secondary 32H30
  • 39A14
  • 35A20