Newton polygons for twisted exponential sums and polynomials P( x d )
Abstract
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 \cite{as1} is tight when
p≡1[de]
, and that it is the actual Newton polygon for any degree
e
polynomial.