Prime and composite Laurent polynomials

Abstract

In 1922 Ritt constructed the theory of functional decompositions of polynomials with complex coefficients. In particular, he described explicitly indecomposable polynomial solutions of the functional equation f(p(z))=g(q(z)). In this paper we study the equation above in the case when f,g,p,q are holomorphic functions on compact Riemann surfaces. We also construct a self-contained theory of functional decompositions of rational functions with at most two poles generalizing the Ritt theory. In particular, we give new proofs of the theorems of Ritt and of the theorem of Bilu and Tichy.

Some of the proofs given in sections 6-8 are simplified. Some other small alterations were made

References in corpus (2)

• Algebraic curves P(x)-Q(y)=0 and functional equations
• On the equation P(f)=Q(g), where P,Q are polynomials and f,g are entire functions

Cited by in corpus (3)

• On Ritt's polynomial decomposition theorems
• On rational functions orthogonal to all powers of a given rational function on a curve
• Analogues of the Jordan-Holder theorem for transitive G-sets