Let Fq denote the finite field of order q and characteristic p. A permutation polynomial (or PP) f(x) ∈ Fq[x] is a bijection of Fq onto itself. If f(x) ∈ Fq is a permutation polynomial over Fq^m  for infinitely many m, then f(x) is said to be an exceptional polynomial over Fq. In general it is not difficult to construct a random PP for a given Fq. Particular simple structures or additional extraordinary properties are usually required in applications of PPs in other areas of mathematics and engineering, such as cryptography, coding theory, or combinatorial designs. Permutation polynomials meeting these criteria are usually difficult to find. The study of permutation polynomials over finite fields is motivated not only by their theoretical importance, but also by their remarkable applications to cryptography, combinatorial designs, and coding theory. A polynomial f(x) ∈ Fq[x] is a complete permutation polynomial (or CPP) of Fq if both f(x) and f(x) + x are permutation polynomials of Fq. CPPs are also related to bent and negabent functions which are studied for a number of applications in cryptography, combinatorial designs, and coding theory. A standard connection between permutation polynomials and algebraic curves over finite fields has been found to be useful to prove non-extince results for both PP and exceptional polynomials.

Permutation trinomials are investigated in [2018/1], [2018/8], [2020/1], [2021/3], [2021/6], [2021/17], whereas in [2018/2] it has been shown that the above mentioned connection can be often used to simplify proofs; see also  [2018/10]. Complete permutation polynomials are studied in [2016/7] and [2017/2], also in connection with exceptional polynomials. The paper [2021/10] deals with non-existence results on permutation rational functions.