Asymptotic variation of L functions of one-variable exponential sums

Fix an integer d ≧ 3. Let double struck A sign^d be the dimension-d affine space over the algebraic closure ℚ̄ of ℚ, identified with the coefficient space of degree-d monic polynomials f(x) in one variable x. For any f(x) in double struck A sign^d(ℚ̄), let ℚ(f) be the field generated by coefficients of f in ℚ̄. For each prime p coprime to d, pick an embedding from ℚ̄ to ℚ̄ _p, once and for all. Let ℘ be the place in ℚ(f) lying over p specified by the embedding of ℚ̄ in ℚ̄_p (with residue field double struck F sign_q say). Suppose f ε double struck A sign^d(ℚ̄ ∩ ℤ̄_p), let NP(f(x) mod ℘) denote the q-adic Newton polygon of the L function L(f(x) mod ℘; T) of exponential sums of f mod ℘. We prove that there is a Zariski dense open subset script U sign defined over ℚ in double struck A sign^d such that for every geometric point f(x) in script U sign(ℚ̄) and p large enough (depending only on f) one has NP(f mod ℘) = GNP(double struck A sign^d; double struck F sign _p) and lim_p→∞ NP(f(x)mod ℘) = HP(double struck A sign^d), where GNP(double struck A sign^d; double struck F sign_p) and HP(double struck A sign^d) are the generic Newton polygon and the Hodge polygon, respectively (see [23]).