In [3], Carlitz determined conditions under which infinite families of polynomials have solutions in a finite field. In this paper we extend

We present an elementary method for evaluating the order of p-divisibility of exponential sums over a prime field. This method unifies and sometimes improves previously known results of Ax-Katz, Moreno-Moreno, Adolphson-Sperber, and Cao-Sun.

In general, the methods to estimate the p-divisibility of exponential sums or the number of solutions of systems of polynomial equations over finite fields are non-elementary. In this paper we present the covering method, an elementary combinatorial method that can be used to compute the exact p-divisibility of exponential sums over a prime field. The results here allow us to compute the exact p-divisibility of exponential sums of new families of polynomials, to unify and improve previously known results, and to construct families of systems of polynomial equations over finite fields that are solvable.

No abstract

Abstract. In this paper we study permutations of finite fields F q that decompose as products of cycles of the same length, and are obtained using monomials. We give the necessary and sufficient conditions on the exponent i to obtain such permutations. We also present formulas for counting the number of this type of permutations. An application to the construction of encoders for turbo codes is also discussed.