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 B1(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 B1(l) terms. Here, we give explicit estimates for this function or, more precicely, we prove that one may take for instance. Moreover, in the case l = 2, a better result is obtained using the same strategy.