Lacunary polynomials and compositions

This thesis deals with lacunary polynomial compositions, that is, polynomial compositions having a fixed number of terms, with an eye towards some arithmetic applications. More specifically, we start by giving some results on polynomial powers having few terms, and then show how these results can be applied to study integer perfect powers having few non-zero digits in their representation in a fixed basis. In relation to this last problem, we also show that, for any fixed basis, there are infinitely many perfect squares having a given number of non-zero digits in their representation, with very few exceptions (which have already been treated in the literature). We then proceed to study lacunary polynomial compositions, focusing on polynomial compositions having relatively many "pure" terms, and apply our results to the study of Universal Hilbert Sets generated by functions associated to linear recurrence relations having only simple roots. After that, we briefly discuss the general case, and provide some evidence towards a general question concerning the minimum number of terms of a composition, in function of the number of variables of the inner polynomial. In the last chapter, we shift our focus towards some additive problems related to the study of these compositions. In particular, we describe a problem concerning additive factorization lengths between terms appearing as exponents of our compositions; then, we study, in the context of additive monoids, an invariant codifying the variations between lengths of additive factorizations of the same integer, showing how these factorizations are very difficult to control even in the most simple cases.