Complete decomposition of Dickson-type polynomials and related Diophantine equations

Abstract We characterize decomposition over C of polynomials f n ( a , B ) ( x ) defined by the generalized Dickson-type recursive relation ( n ⩾ 1 ) f 0 ( a , B ) ( x ) = B , f 1 ( a , B ) ( x ) = x , f n + 1 ( a , B ) ( x ) = x f n ( a , B ) ( x ) − a f n − 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 equation f n ( a , B ) ( x ) = f m ( a ˆ , B ˆ ) ( y ) . This extends and completes work of Dujella/Tichy and Dujella/Gusic 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.

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