Explicit evaluations of some Weil sums

where f ∈ Fq[X ]. These sums are also known as Weil sums. The problem of explicitly evaluating these sums is quite often difficult. Results giving estimates for the absolute value of the sum are more common and such results have been regularly appearing for many years. The book [5] by Lidl and Niederreiter gives an overview of this area of research in the concluding remarks of Chapter 5. As with previous explicit evaluations the special form of our polynomial will play an integral part. In [1] Carlitz obtained explicit evaluations of Weil sums with f(X) = aX + bX . His methods involved first obtaining evaluations when b = 0 and then proceeding to the general case. This article is largely a generalisation of the methods used by Carlitz in the first part of [1]. A further article dealing with the second part of Carlitz’ evaluation method is under preparation. The polynomials studied in this article are of the form f(X) = aX +1 where α is an arbitrary natural number. These monomials are a subset of the class of polynomials known as Dembowski-Ostrom polynomials (or D-O polynomials). We may define a D-O polynomial to be any polynomial which, when reduced, has the shape