Quasi-polynomial

In mathematics, a quasi-polynomial (pseudo-polynomial) is a generalization of polynomials. While the coefficients of a polynomial come from a ring, the coefficients of quasi-polynomials are instead periodic functions with integral period. Quasi-polynomials appear throughout much of combinatorics as the enumerators for various objects.

A quasi-polynomial can be written as $$q(k) = c_d(k) k^d + c_{d-1}(k) k^{d-1} + \cdots + c_0(k)$$, where $$c_i(k)$$ is a periodic function with integral period. If $$c_d(k)$$ is not identically zero, then the degree of $$q$$ is $$d$$. Equivalently, a function $$f \colon \mathbb{N} \to \mathbb{N}$$ is a quasi-polynomial if there exist polynomials $$p_0, \dots, p_{s-1}$$ such that $$f(n) = p_i(n)$$ when $$i \equiv n \bmod s$$. The polynomials $$p_i$$ are called the constituents of $$f$$.

Examples

 * Given a $$d$$-dimensional polytope $$P$$ with rational vertices $$v_1,\dots,v_n$$, define $$tP$$ to be the convex hull of $$tv_1,\dots,tv_n$$. The function $$L(P,t) = \#(tP \cap \mathbb{Z}^d)$$ is a quasi-polynomial in $$t$$ of degree $$d$$. In this case, $$L(P,t)$$ is a function $$\mathbb{N} \to \mathbb{N}$$. This is known as the Ehrhart quasi-polynomial, named after Eugène Ehrhart.
 * Given two quasi-polynomials $$F$$ and $$G$$, the convolution of $$F$$ and $$G$$ is
 * $$(F*G)(k) = \sum_{m=0}^k F(m)G(k-m)$$
 * which is a quasi-polynomial with degree $$\le \deg F + \deg G + 1.$$