  Let $f(x_1,...,x_k)$ be a polynomial over a field $K$. This paper considers
such questions as the enumeration of the number of nonzero coefficients of $f$
or of the number of coefficients equal to $\alpha\in K^*$. For instance, if
$K=\ff_q$ then a matrix formula is obtained for the number of coefficients of
$f^n$ that are equal to $\alpha\in \ff_q^*$, as a function of $n$. Many
additional results are obtained related to such areas as lattice path
enumeration and the enumeration of integer points in convex polytopes.
