Restrictions on meromorphic solutions of Fermat type equations

Abstract

The Fermat type functional equations $(*)\, f_1^n+f_2^n+\cdots +f_k^n=1$ , where n and k are positive integers, are considered in the complex plane. Our focus is on equations of the form (*) where it is not known whether there exist non-constant solutions in one or more of the following four classes of functions: meromorphic functions, rational functions, entire functions, polynomials. For such equations, we obtain estimates on Nevanlinna functions that transcendental solutions of (*) would have to satisfy, as well as analogous estimates for non-constant rational solutions. As an application, it is shown that transcendental entire solutions of (*) when n = k(k − 1) with k ≥ 3, would have to satisfy a certain differential equation, which is a generalization of the known result when k = 3. Alternative proofs for the known non-existence theorems for entire and polynomial solutions of (*) are given. Moreover, some restrictions on degrees of polynomial solutions are discussed.

Difference radical in terms of shifting zero and applications to the Stothers-Mason theorem

“…Example 2.2 Consider a polynomial f (z) = z 2 (z − 1) 3 . It gives us that z = 0 is a zero of f with order 2, and z = 1 is a zero of f with order 3.…”

Entire solutions of several quadratic binomial and trinomial partial differential-difference equations in $$ {\mathbb {C}}^2 $$

“…In general, it is difficult to find entire and meromorphic solutions for a nonlinear PDE. By employing Nevanlinna theory and the method of complex analysis, there were a number of literature focusing on the solutions of some PDEs and theirs many variants, readers can refer to [3,4,11,16,17,22,24,29,35].…”

Transcendental entire solutions of quadratic functional equations

“…and proved that equation (1.2) does not admit any transcendental meromorphic solution. For different aspects of the solutions of Fermat-type functional equations including some recent developments, we refer to the articles [2,5,10,12,16,24,27,37,40,47] and references therein.…”

“…In general, it is difficult to find entire and meromorphic solutions for a nonlinear PDE. By employing Nevanlinna theory and the method of complex analysis, there were a number of literature focusing on the solutions of some PDEs and their many variants, readers can refer to [6,15,21,25,38]. The solutions of Fermat-type P DEs were investigated by [18,23,36].…”