p-Density, exponential sums and Artin-Schreier curves

In this paper we define the $p$-density of a finite subset $D\subset \text{ma}{N}^r$, and show that it gives a good lower bound for the $p$-adic valuation of exponential sums over finite fields of characteristic $p$. We also give an application: when $r=1$, the $p$-density is the first slope of the generic Newton polygon of the family of Artin-Schreier curves associated to polynomials with their exponents in $D$.