  Let $p$ denote an odd prime. In this paper, we are concerned with the
$p$-divisibility of additive exponential sums associated to one variable
polynomials over a finite field of characteristic $p$, and with (the very close
question of) determining the Newton polygons of some families of Artin-Schreier
curves, i.e. $p$-cyclic coverings of the projective line in characteristic $p$.
  We first give a lower bound on the $p$-divisibility of exponential sums
associated to polynomials of fixed degree. Then we show that an Artin-Schreier
curve defined over a finite field of characteristic $p$ cannot be supersingular
when its genus $g$ has the form $(p-1)\left(i(p^n-1)-1\right)/2$ for some
$1\leq i\leq p-1$ and $n\geq 1$ such that $n(p-1)>2$. We also determine the
first vertex of the generic Newton polygon of the family of $p$-rank $0$
Artin-Schreier curves of fixed genus, and the associated Hasse polynomial.
