Several classes of permutation polynomials and their differential uniformity properties

The notion of $c$-differential uniformity has recently received a lot of popularity because of its potential applications in cryptography (we point out that a connection with difference sets in some quasigroups is already realized for perfect $c$-nonlinear functions, in a recent manuscript~\cite{AMS22}). The construction of functions, especially permutations, with low $c$-differential uniformity is an interesting problem in this area, and recent work has focused heavily in this direction. We provide a few classes of permutation polynomials with low $c$-differential uniformity. The used technique involves handling various Weil sums, as well as analyzing some equations in finite fields, and we believe these can be of independent interest.