m at h . N T ] 1 8 Ju n 20 07 NEWTON POLYGONS FOR TWISTED EXPONENTIAL SUMS AND POLYNOMIALS

We study the p-adic absolute value of the roots of the L-functions associated to certain twisted character sums, and additive character sums associated to polynomials P (x d), when P varies among the space of polynomial of fixed degree e over a finite field of characteristic p. For sufficiently large p, we determine in both cases generic Newton polygons for these L-functions, which is a lower bound for the Newton polygons, and the set of polynomials of degree e for which this generic polygon is attained. In the case of twisted sums, we show that the lower polygon defined in [1] is tight when p ≡ 1 [de], and that it is the actual Newton polygon for any degree e polynomial.