# On Certain Fermat Diophantine Functional Equations in C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackag ...

On Certain Fermat Diophantine Functional Equations in C2\documentclass[12pt]{minimal}... In this paper, we study entire solutions and meromorphic solutions of the following Fermat Diophantine functional equations hz1,z2f+kz1,z2gn=1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}\begin{aligned} h\left( z_{1}, z_{2}\right) f+k\left( z_{1}, z_{2}\right) g^{n}=1 \end{aligned}\end{document}in C2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {C}^{2}$$\end{document} for integers n≥2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n\ge 2$$\end{document}, where hz1,z2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$h\left( z_{1}, z_{2}\right)$$\end{document} and kz1,z2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$k\left( z_{1}, z_{2}\right)$$\end{document} are non-zero meromorphic functions in C2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {C}^{2}$$\end{document}, and show that f and g can reduce to a constant or rational function under the conditions that (kgn)z1≢0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(kg^n)_{z_1}\not \equiv 0$$\end{document}, L(hf)z2⊆L(kgn)z1/(kgn-1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {L}}\left( (hf)_{z_2}\right) \subseteq {\mathcal {L}}\left( (kg^n)_{z_1}/(kg^{n-1})\right)$$\end{document} and L(f)⊆L(kgn)z1/(kgn-1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {L}}(f)\subseteq {\mathcal {L}}\left( (kg^n)_{z_1}/(kg^{n-1})\right)$$\end{document} ignoring multiplicities or counting multiplicities. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Computational Methods and Function Theory Springer Journals

# On Certain Fermat Diophantine Functional Equations in C2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackag ...

, Volume OnlineFirst – May 16, 2022
14 pages

/lp/springer-journals/on-certain-fermat-diophantine-functional-equations-in-c2-documentclass-6UOyDffhAi
Publisher
Springer Journals
Copyright © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2022
ISSN
1617-9447
eISSN
2195-3724
DOI
10.1007/s40315-022-00450-8
Publisher site
See Article on Publisher Site

### Abstract

In this paper, we study entire solutions and meromorphic solutions of the following Fermat Diophantine functional equations hz1,z2f+kz1,z2gn=1\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}\begin{aligned} h\left( z_{1}, z_{2}\right) f+k\left( z_{1}, z_{2}\right) g^{n}=1 \end{aligned}\end{document}in C2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {C}^{2}$$\end{document} for integers n≥2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n\ge 2$$\end{document}, where hz1,z2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$h\left( z_{1}, z_{2}\right)$$\end{document} and kz1,z2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$k\left( z_{1}, z_{2}\right)$$\end{document} are non-zero meromorphic functions in C2\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\mathbb {C}^{2}$$\end{document}, and show that f and g can reduce to a constant or rational function under the conditions that (kgn)z1≢0\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$(kg^n)_{z_1}\not \equiv 0$$\end{document}, L(hf)z2⊆L(kgn)z1/(kgn-1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {L}}\left( (hf)_{z_2}\right) \subseteq {\mathcal {L}}\left( (kg^n)_{z_1}/(kg^{n-1})\right)$$\end{document} and L(f)⊆L(kgn)z1/(kgn-1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {L}}(f)\subseteq {\mathcal {L}}\left( (kg^n)_{z_1}/(kg^{n-1})\right)$$\end{document} ignoring multiplicities or counting multiplicities.

### Journal

Computational Methods and Function TheorySpringer Journals

Published: May 16, 2022

Keywords: Entire solution; Meromorphic solutions; Fermat type functional equations; Nevanlinna theory; 32A15; 32A22; 35F20

### References

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