  For an odd prime $p$ and an integer $w\ge 1$, polynomial quotients
$q_{p,w}(u)$ are defined by $$ q_{p,w}(u)\equiv \frac{u^w-u^{wp}}{p} \bmod p ~~
\mathrm{with}~~ 0 \le q_{p,w}(u) \le p-1, ~~u\ge 0, $$ which are
generalizations of Fermat quotients $q_{p,p-1}(u)$.
  First, we estimate the number of elements $1\le u<N\le p$ for which
$f(u)\equiv q_{p,w}(u) \bmod p$ for a given polynomial $f(x)$ over the finite
field $\mathbb{F}_p$. In particular, for the case $f(x)=x$ we get bounds on the
number of fixed points of polynomial quotients.
  Second, before we study the problem of estimating the smallest number (called
the Waring number) of summands needed to express each element of $\mathbb{F}_p$
as sum of values of polynomial quotients, we prove some lower bounds on the
size of their value sets, and then we apply these lower bounds to prove some
bounds on the Waring number using results from bounds on additive character
sums and additive number theory.
