Hasse-Weil type theorems and relevant classes of polynomial functions

Several types of functions over  nite  elds have relevant applications in applied areas of mathematics, such as cryptography and coding theory. Among them, planar functions, APN permutations, permutation polynomials, and scattered polynomials have been widely studied in the last few years. In order to provide both non-existence results and explicit constructions of in nite families, sometimes algebraic varieties over  nite  elds turn out to be a useful tool. In a typical argument involving algebraic varieties, the key step is estimating the number of their rational points over some  nite  eld. For this reason, Hasse-Weil type theorems (such as Lang-Weil's and Serre's) play a fundamental role.