POLYNOMIALS FOR HYPER-KLOOSTERMAN SUMS

Abstract

Fix an integer m > 1, and let [Formula: see text] denote the group of reduced residues modulo m and set ζm = exp (2 πi/m). Let [Formula: see text] denote the multiplicative inverse of x modulo m. The n-dimensional Kloosterman sums are defined by [Formula: see text] and satisfy the polynomial [Formula: see text] where the product is taken over a reduced system of residues modulo m. Here we give a natural factorization of fm(x); namely, [Formula: see text] where σ runs through the n + 1-st power classes modulo m. Questions concerning the determination of the factors [Formula: see text] (or at least their beginning coefficients), their reducibility over the rational field Q and duplications among the factors are studied. The treatment generalizes results of the author in the classical case n = 1, and relies heavily on the recent explicit evaluation of hyper-Kloosterman sums for prime powers.