Explicit bounds for composite lacunary polynomials

Abstract

Let f, g, h ∈ C [x] be non-constant complex polynomials satisfying f (x) = g(h(x)) and let f be lacunary in the sense that it has at most l non-constant terms. Zannier proved in [9] that there exists a function B 1 (l) on N, depending only on l and with the property that h(x) can be written as the ratio of two polynomials having each at most B 1 (l) terms. Here, we give explicit estimates for this function or, more precisely, we prove that one may take for instanceMoreover, in the case l = 2, a better bound is obtained using the same strategy.