In 1922 Ritt described polynomial solutions of the functional equation $P(f) = Q(g).$ In this paper we describe solutions of the equation above in the case when $P, Q$ are polynomials while $f, g$ are allowed to be arbitrary entire functions. In fact, we describe solutions of the more general functional equation $s = P(f) = Q(g),$ where $s, f, g$ are entire functions and $P, Q$ are arbitrary rational functions. As an application we solve the problem of description of "strong uniqueness polynomials" for entire functions.

https://www.math.bgu.ac.il/~pakovich/Publications/e.pdf

On the equation P(f) = Q(g), where P, Q are polynomials and f, g are entire functions

In this paper we give several conditions implying the irreducibility of the algebraic curve P(x)-Q(y)=0, where P,Q are rational functions. We also apply the results obtained to the functional equations P(f)=Q(g) and P(f)=cP(g), where c\in C. For example, we show that for a generic pair of rational functions P,Q the first equation has no non-constant solutions f,g meromorphic on C whenever (°P-1)(°Q-1) \geq 2.

Algebraic curves P(x)-Q(y)=0 and functional equations

Let E be an algebraically closed field of characteristic 0 which is either C or a complete ultrametric field K. We consider the composition of meromorphic functions h ◦ f where h is meromorphic in all E and f is meromorphic either in E or in an open disk of K. We then look for a condition on h in order that if 2 similar functions f, g satisfy h ◦ f( am) = h ◦ g(am) where (am) is a bounded sequence satisfying certain condition, this implies f = g. Particularly we generalize to meromorphic functions previous results

https://hal.archives-ouvertes.fr/hal-00681100/document

Meromorphic functions of uniqueness

In this article, we give several conditions implying the irreducibility of the algebraic curve P(x) − Q(y) = 0, where P, Q are rational functions. We also apply the results obtained to the functional equations P(f) = Q(g) and P(f) = cP(g), where c ∈ ℂ. For example, we show that for a generic pair of rational functions P, Q the first equation has no non-constant solutions f, g meromorphic on ℂ whenever (deg P − 1)(deg Q − 1) ≥ 2.

Algebraic curves P(x) − Q(y) = 0 and functional equations

A finite set S in the extended complex plane is called a unique range set counting multiplicities (URSCM) (unique range set ignoring multiplicities, URSIM) for meromorphic functions on if any two nonconstant meromorphic functions f and g in satisfying counting multiplicities (CM) (ignoring multiplicities IM), then f = g. The question ‘What is the smallest cardinality for such a set?’ is still a challenging and open one. In this article, for meromorphic functions on we exhibit a URSCM and a URSIM of 13 and 19 elements, respectively. We also characterize a unique range set for entire functions of order <1 on . Some conjectures are proposed for further studies. §Dedicated to Professor Guochun Wen on the occasion of his 75th birthday.

Unique range sets for meromorphic functions on

We prove that for a generic pair (P, Q) of polynomials P of degree n and Q of degree m, where m, n are satisfying some conditions, P(f)=Q(g) for meromorphic functions f,g implies f=const, g=const. We also give another proof of the statement saying that a generic polynomial of degree at least 5 is a uniqueness polynomial for meromorphic functions.

Hui Khoai Ha

Papers

On the Functional Equation P(F)=Q(G)