Polynomial Coefficient Enumeration

TL;DR

This paper investigates the enumeration of polynomial coefficients over fields, providing formulas for counting specific coefficient types in polynomial powers and exploring related combinatorial enumeration problems.

Contribution

It introduces new formulas for counting coefficients of polynomial powers over finite fields and connects these results to lattice path and convex polytope enumeration.

Findings

Matrix formula for coefficients of polynomial powers over finite fields

Enumeration results linking polynomials to lattice paths

Connections between polynomial coefficients and convex polytope points

Abstract

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.