p-adic variation of L-functions of exponential sums, I

For a polynomial $f(x)$ in $({\mathbb{Z}}_p\cap \mathbb{Q})[x]$ of degree $d>2$ let $L(fp;T)$ be the $L$-function of the exponential sum of $fp$. Let $\mathrm{N}\mathrm{P}(fp)$ denote the Newton polygon of $L(fp;T)$. Let $\mathrm{H}\mathrm{P}(f)$ denote the Hodge polygon of $f$, which is the lower convex hull in the real plane of the points $(n,n(n+1)\mathrm{/}(2d))$ for $0\le n\le d-1$. We prove that there is a Zariski dense subset $\mathcal{U}$ defined over $\mathbb{Q}$ in the space ${\mathbb{A}}^d$ of degree-$d$ monic polynomials over $\mathbb{Q}$ such that for all $f$ in $\mathcal{U}(\mathbb{Q})$ we have ${\lim }_{p\to \mathrm{\infty }}\mathrm{N}\mathrm{P}(fp)=\mathrm{H}\mathrm{P}(f)$. Moreover, we determine the $p$-adic valuation of every coefficient of $L(fp;T)$ for $p$ large enough and $f$ in $\mathcal{U}(\mathbb{Q})$, and that of $L(x^d+a x \bmod p;T)$forall$a\neq 0$.