On lacunary polynomials and a generalization of Schinzel's conjecture

Some interesting questions can be posed regarding the maximum number of terms of a polynomial when dealing with particular operations: for example, Renyi and Erdős asked whether there is a bound on the number of terms of h(x) depending only on the number of terms of h(x)^2. In the last decade, positive answers have been found for very general situations: a conjecture by Schinzel on the case of g(h(x)) having few terms for some complex polynomial g has been proven in [8], and an even more general case where h(x) satisfies F(x,h(x))=0 for some complex polynomial F in two variables has been proven in [2]; moreover, the bounds obtained are dependent very poorly on g and F. We are exposing here the proof of Schinzel’s conjecture contained in [8] and of its aforementioned generalized form contained in [2]; we also give explicit formulas and procedures to calculate the bounds themselves, which were lacking in the original papers.