Complete decomposition of Dickson-type polynomials and related Diophantine equations

We characterize decomposition over C of polynomials fn(a,B)(x) defined by the generalized Dickson-type recursive relation (n⩾1)f0(a,B)(x)=B,f1(a,B)(x)=x,fn+1(a,B)(x)=xfn(a,B)(x)−afn−1(a,B)(x), where B,a∈Q or R. As a direct application of the uniform decomposition result, we fully settle the finiteness problem for the Diophantine equationfn(a,B)(x)=fm(aˆ,Bˆ)(y). This extends and completes work of Dujella/Tichy and Dujella/Gusić concerning Dickson polynomials of the second kind. The method of the proof involves a new sufficient criterion for indecomposability of polynomials with fixed degree of the right component.