Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 40, 2015, 907-921

ON THE EXISTENCE OF SOLUTIONS OF A FERMAT-TYPE DIFFERENCE EQUATION

Nan Li

Shandong University, School of Mathematics, Jinan, Shandong, 250100, P.R. China
and University of Eastern Finland, Department of Physics and Mathematics
P.O. Box 111, 80101 Joensuu, Finland; nanli32787310 'at' 163.com

Abstract. The analogue of Fermat's last theorem for function fields has been investigated by many scholars recently, and Gundersen-Hayman [6] collected the best lower estimates that are known for F_C(n), where F_C(n) is the smallest positive integer k such that the equation

f₁ⁿ + f₂ⁿ + ... + f_kⁿ = 1

has a solution consisting of k nonconstant functions f₁, f₂,..., f_k in C, and C is the ring of meromorphic functions M, rational functions R, entire functions E or polynomials P, respectively. In this paper, we investigate a difference analogue of this problem for the rings of M, R, E, P with certain conditions, and obtain lower bounds for G_C, where G_C(n) is the smallest positive integer k such that the equation

f₁(z)f₁(z + c) ··· f₁(z + (n - 1)c) + ... + f_k(z)f_k(z + c) ··· f_k(z + (n - 1)c) = 1

has a solution consisting of k nonconstant functions f₁, f₂,..., f_k in C.

2010 Mathematics Subject Classification: Primary 30D35; Secondary 39A10.

Key words: Entire functions, meromorphic functions, Nevanlinna theory, Fermat-type equations.

Reference to this article: N. Li: On the existence of solutions of a Fermat-type difference equation. Ann. Acad. Sci. Fenn. Math. 40 (2015), 907-921.

Full document as PDF file

doi:10.5186/aasfm.2015.4051