Relevant functions over finite fields and algebraic varieties

The crux of this thesis lies in the exploration of three families of functions over finite fields: permutation polynomials, almost perfect nonlinear (APN) functions, and scattered polynomials. A unifying factor among these three families of functions is the ability to associate each of them with an appropriate algebraic variety (a curve in the case of permutation and scattered polynomials, or a surface in the case of APN functions) that fully describes them. In all three instances, the properties of such functions can be translated in terms of the existence of specific points, with coordinates over a finite field $\mathbb{F}_{q^n}$, of the respective varieties under consideration. The explicit connections manifest in the following manner: a polynomial is a permutation/APN/scattered if and only if a suitable variety (linked to the starting polynomial) exclusively has $\mathbb{F}_{q^n}$-rational points of a certain shape. Leveraging this linkage alongside Hasse-Weil or Lang-Weil type theorems, which estimate the number of $\mathbb{F}_{q^n}$-rational points a given variety may possess, has yielded numerous results concerning the existence and/or classification of permutation/APN/scattered polynomials.