Diophantine equations with Euler polynomials

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In this paper we determine possible decompositions of Euler polynomials Eₖ(x), i.e. possible ways of writing Euler polynomials as a functional composition of polynomials of lower degree. Using this result together with the well-known criterion of Bilu and Tichy, we prove that the Diophantine equation -1ᵏ +2 ᵏ - + (-1)ˣ xᵏ=g(y), with g∈ Q[X] of degree at least 2 and k≥ 7, has only finitely many integers solutions x, y unless polynomial g can be decomposed in ways that we list explicitly.