Polynomial Coefficient Enumeration

Let $f(x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_k)$ 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 $\alpha \in K^{\ast }$. For instance, if $K={\text{ff}}_q$ then a matrix formula is obtained for the number of coefficients of $f^n$ that are equal to $\alpha \in {\text{ff}}_q^{\ast }$, 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.