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

  • Ling Xu
  • Tingbin CaoEmail author


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.


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

Mathematics Subject Classification

Primary 39A45 Secondary 32H30 39A14 35A20 



  1. 1.
    Adams, C.R.: On the linear ordinary \(q\)-difference equation. Ann. Math. 30(1/4), 195–205 (1928–1929)Google Scholar
  2. 2.
    Biancofiore, A., Stoll, W.: Another proof of the lemma of the logarithmic derivative in several complex variables. In: Fornaess, J. (ed.) Recent developments in several complex variables, pp. 29–45. Princeton University Press, Princeton (1981)CrossRefGoogle Scholar
  3. 3.
    Cao, T.B., Korhonen, R.J.: A new version of the second main theorem for meromorphic mappings intersecting hyperplanes in several complex variables. J. Math. Anal. Appl. 444(2), 1114–1132 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Chen, Z.X., Shon, K.H.: Estimates for the zeros of differences of meromorphic functions. Sci. China Ser. A Math. 52(11), 2447–2458 (2009)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Chiang, Y.M., Feng, S.J.: On the Nevanlinna characteristic of \(f(z+\eta )\) and difference equations in the complex plane. Ramanujan J. 16(1), 105–129 (2008)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Gross, F.: On the equation \(f^{n}+g^{n}=1,\). Bull. Am. Math. Soc. 72, 86–88 (1966)CrossRefGoogle Scholar
  7. 7.
    Halburd, R.G., Korhonen, R.J.: Difference analogue of the lemma on the logarithmic derivative with applications to difference equations. J. Math. Anal. Appl. 314, 477–487 (2006)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Halburd, R.G., Korhonen, R.J.: Nevanlinna theory for the difference operator. Ann. Acad. Sci. Fenn. Math. 31, 463–478 (2006)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Halburd, R.G., Korhonen, R.J.: Finite-order meromorphic solutions and the discrete Painleve equations. Proc. Lond. Math. Soc. 94(2), 443–474 (2007)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Hu, P.C., Yang, C.C.: The Tumura–Clunie theorem in several complex variables. Bull. Aust. Math. Soc. 90, 444–456 (2014)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Hu, P.C.: Malmquist type theorem and factorization of meromorphic solutions of partial differential equations. Complex Var. 27, 269–285 (1995)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Hu, P.C., Li, P., Yang, C.C.: Unicity of Meromorphic Mappings, Advances in Complex Analysis and its Applications, vol. 1. Kluwer Academic Publishers, Dordrecht, Boston, London (2003)Google Scholar
  13. 13.
    Khavinson, D.: A note on entire solutions of the eiconal equation. Am. Math. Mon. 102, 159–161 (1995)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Korhonen, R.J.: A difference Picard theorem for meromorphic functions of several variables. Comput. Methods Funct. Theory 12(1), 343–361 (2012)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Lelong, P.: Fonctionnelles Analytiques et Fonctions Entières (n variables). Presses de L’Université de Montréal (1968)Google Scholar
  16. 16.
    Li, B.Q.: On entire solutions of Fermat type partial differential equations. Int. J. Math. 15, 473–485 (2004)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Li, B.Q.: On meromorphic solutions of \(f^{2}+g^{2}=1,\). Math. Z. 258(4), 763–771 (2008)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Li, B.Q.: On reduction of functional-differential equations. Complex Var. 31, 311–324 (1996)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Liu, K., Cao, T.B., Cao, H.Z.: Entire solutions of Fermat-type differential-difference equations. Arch. Math. 99, 147–155 (2012)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Liu, K., Yang, L.Z.: A note on meromorphic solutions of Fermat-types equations. An. Stiint. Univ. Al. I. Cuza Lasi Mat. (N. S.) 1, 317–325 (2016)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Lü, F., Han, Q.: On the Fermat-type equation \(f^{3}(z)+f^{3}(z+c)=1,\). Aequat. Math. 91, 129–136 (2017)CrossRefGoogle Scholar
  22. 22.
    Montel, P.: Lecons sur les familles normales de fonctions analytiques et leurs applications, pp. 135–136. Gauthier-Villars, Paris (1927)zbMATHGoogle Scholar
  23. 23.
    Pólya, G.: On an integral function of an integral function. J. Lond. Math. Soc. 1, 12–15 (1926)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Ronkin, L.I.: Introduction to the Theory of Entire Functions of Several Variables, Moscow: Nauka 1971 (Russian). American Mathematical Society, Providence (1974)Google Scholar
  25. 25.
    Saleeby, E.G.: Entire and meromorphic solutions of Fermat type partial differential equations. Analysis 19, 69–376 (1999)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Saleeby, E.G.: On entire and meromorphic solutions of \(\lambda u^{k}+\sum _{i=1}^{n}u_{z_{i}}^{m}=1,\). Complex Var. 49(3), 101–107 (2004)MathSciNetGoogle Scholar
  27. 27.
    Shabat, B.V.: Functions of Several Variables, Introduction to Complex Analysis, Part II, Translation Mathematical Monographs, vol. 110. American Mathematical Society, Providence, RI (1992)Google Scholar
  28. 28.
    Stoll, W.: Holomorphic Functions of Finite Order in Several Complex Variables. American Mathematical Society, Providence (1974)Google Scholar
  29. 29.
    Taylor, R., Wiles, A.: Ring-theoretic properties of certain Hecke algebra. Ann. Math. 141, 553–572 (1995)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Wiles, A.: Modular elliptic curves and Fermats last theorem. Ann. Math. 141, 443–551 (1995)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Ye, Z.: On Nevanlinna’s second main theorem in projective space. Invent. Math. 122, 475–507 (1995)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsNanchang UniversityNanchangChina

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