2025/09/02 by Tingbin Cao, Jun Wang, Cao, Tingbin +3
Mathematics · Computer Science · #Meromorphic and Entire Functions #Polynomial and algebraic computation #Advanced Differential Equations and Dynamical Systems
paper · pdf · doi:10.48550/arxiv.2509.01862
Assume that n is a positive integer, pj (j=1,2, ⋯, 6) are polynomials, p is an irreducible polynomial, and f is an entire function on ℂn. Let L(f)=∑j=1s qtjf_ztj and f(z)=f(z1+c1, …, zn+cn), where qtj (j=1,2, ⋯, s≤ n) are non-zero polynomials on ℂn and c=(c1, …, cn)∈ ℂn∖\0\. We show the structures of all entire solutions to the non-linear partial differential-difference equation (p1 L(f)+p2f+p5 f)2+(p3L(f)+p4f+p6 f)2=p. The partial differential-difference equation is called a Fermat-type partial differential-difference equation (PDDE). Further, we find many sufficient conditions and/or necessary conditions for the existence, as well as the concrete representations, of entire solutions to the Fermat-type PDDE. We also demonstrate several examples on ℂ2 with non-constant coefficients to verify that all representations in our theorems exist and are accurate and that the entire solutions to the Fermat-type PDDEs could have finite or infinite growth order. Our theorems unify and extend previous results (see, e.g., [2, 3, 10, 12, 32]).