  In this paper we prove that there exists a Zariski dense open subset U
defined over the rationals Q in the space of all one-variable rational
functions with arbitrary k poles of prescribed orders, such that for every
geometric point f in U(Qbar)$, the L-function of the exponential sum of f at a
prime p has Newton polygon approaching the Hodge polygon as p approaches
infinity. As an application to algebraic geometry, we prove that the p-adic
Newton polygon of the zeta function of a p-cover of the projective line totally
ramified at arbitrary k points of prescribed orders has an asymptotic generic
lower bound.

Number Theory;Algebraic Geometry