Quasi-polynomial

For quasi-polynomial bounds in the analysis of algorithms, see Quasi-polynomial growth. For functions that are characteristic-functions of linear time-delay systems, see Delay differential equation § The characteristic equation.

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 ${\displaystyle q(k)=c_{d}(k)k^{d}+c_{d-1}(k)k^{d-1}+\cdots +c_{0}(k)}$, where ${\displaystyle c_{i}(k)}$ is a periodic function with integral period. If ${\displaystyle c_{d}(k)}$ is not identically zero, then the degree of ${\displaystyle q}$ is ${\displaystyle d}$. Equivalently, a function ${\displaystyle f\colon \mathbb {N} \to \mathbb {N} }$ is a quasi-polynomial if there exist polynomials ${\displaystyle p_{0},\dots ,p_{s-1}}$ such that ${\displaystyle f(n)=p_{i}(n)}$ when ${\displaystyle i\equiv n{\bmod {s}}}$. The polynomials ${\displaystyle p_{i}}$ are called the constituents of ${\displaystyle f}$.

Examples

• Given a ${\displaystyle d}$ -dimensional polytope ${\displaystyle P}$  with rational vertices ${\displaystyle v_{1},\dots ,v_{n}}$ , define ${\displaystyle tP}$  to be the convex hull of ${\displaystyle tv_{1},\dots ,tv_{n}}$ . The function ${\displaystyle L(P,t)=\#(tP\cap \mathbb {Z} ^{d})}$  is a quasi-polynomial in ${\displaystyle t}$  of degree ${\displaystyle d}$ . In this case, ${\displaystyle L(P,t)}$  is a function ${\displaystyle \mathbb {N} \to \mathbb {N} }$ . This is known as the Ehrhart quasi-polynomial, named after Eugène Ehrhart.
• Given two quasi-polynomials ${\displaystyle F}$  and ${\displaystyle G}$ , the convolution of ${\displaystyle F}$  and ${\displaystyle G}$  is

${\displaystyle (F*G)(k)=\sum _{m=0}^{k}F(m)G(k-m)}$ 

which is a quasi-polynomial with degree ${\displaystyle \leq \deg F+\deg G+1.}$ 

References

• Stanley, Richard P. (1997). Enumerative Combinatorics, Volume 1. Cambridge University Press. ISBN 0-521-55309-1, 0-521-56069-1.