On decompositions of binary recurrent polynomials

Polynomial decomposition expresses a polynomial f as the functional composition f= g∘ h of lower degree polynomials g and h, and has various applications. In this paper, we will show that for a minimal, non-degenerate, simple, binary, linearly recurrent sequence (Gn(x))n=0∞ of complex polynomials whose coefficients in the Binet form are constants, if G _n(x) = g(h(x)) , then apart from some exceptional situations that have to be taken into account, the degree of g is bounded by a constant independent of n. We will build on a general but conditional result of this type that already exists in the literature. We will then present one Diophantine application of the main result.