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 $q(k) = c_d(k) k^d + c_(k) k^ + \cdots + c_0(k)$, 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 $f \colon \mathbb \to \mathbb$ is a quasi-polynomial if there exist polynomials $p_0, \dots, p_$ 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 $L(P,t) = \#(tP \cap \mathbb^d)$ is a quasi-polynomial in $t$ of degree $d$. In this case, $L(P,t)$ is a function $\mathbb \to \mathbb$. 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
:: $(F*G)(k) = \sum_^k F(m)G(k-m)$
which is a quasi-polynomial with degree $\le \deg F + \deg G + 1.$

See also

* Ehrhart polynomial

References

* Stanley, Richard P. (1997)
''Enumerative Combinatorics'', Volume 1
Cambridge University Press. , 0-521-56069-1.

Polynomials
Algebraic combinatorics

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