Generic A-family of exponential sums

Let s→:=(s1,s2,...,sm) with _s1 < ≡ < _{s m} being positive integers. Let A(s→) be the space of all 1-variable polynomials f(x)=∑ℓ=1maℓxsℓ parameterized by coefficients a→=(a1,...,am) with _{a m} ≠ 0. We study the p-adic valuation of the roots of the L-function of exponential sum of f- for modulo p reduction of any generic point f∈A(s→)(Q-). Let NP(f-) be the normalized p-adic Newton polygon of the L function of exponential sums of f- Let GNP(A(s→),F-p) be the generic Newton polygon for A(s→) over F-p, and let HP(A(s→)):=NPp(∏i=1d-1(1-pidT)) be the absolute lower bound of NP(A(s→)). One knows that NP(f-)≺GNP(A(s→);F-p)≺HP(f-) for all prime p and for all f-∈A(s→)(Q-), and these equalities hold only when p≡ 1 mod d. In the case s→=(s,d) with s < d coprime we provide a computational method to determine GNP(A(s,d),F-p) explicitly by constructing its generating polynomial _{H r} ∈ Q[_{X r,1}, _{X r,2}, ..., _{X r,d -1}] for each residue class p≡ r mod d. For p≡ r mod d (with 2 ≤ r ≤ d - 1 coprime to d) large enough Hr=∑n=1d-1hr,n,kr,nXr,nkr,n with ∏n=1d-1hr,n,kr,n≠0 if and only if GNP(A(s,d),F-p) has its breaking points after the origin at((n,n(n+1)2d+(1-sd)kr,np-1))n=1,2,...,d-1. If a ≠ 0 then for any f=xd+axs∈A(s,d)(Q-) and for any prime p≡ r mod d large enough we have that NP(f-)=GNP(A(s,d),F-p) andlimp→∞NP(f-)=HP(A(s,d)).Our method applies to compute the generic Newton polygon of Artin-Schreier family ^{y p} - y = ^{x d} + a^{x s} parameterized by a for p large enough.