Quasipolynomial

example

• assume $a_n = \left((-1)^n+1\right)+\left((-1)^n+3\right) n$
• then

$$ \sum_{n=0}^{\infty}a_nt^n = \frac{2 \left(t^3+3 t^2+t+1\right)}{(1-t)^2 (t+1)^2} $$

some results

thm (Ehrhart's theorem for rational polytopes)

If $P$ is a rational convex $d$-polytope, then $L_{P}(t)$ is a quasipolynomial in $t$ of degree $d$. Its period divides the least common multiple of the denominator of the coordinates of the vertices of $P$.

lemma (Beck-Robins ex. 3.19)

If $\sum_{t \ge 0} f(t)z^t = \frac{g(z)}{h(z)}$, then $f$ is a quasipolynomial of degree $d$ with period $p$ if and only if $g$ and $h$ are polynomials such that $\deg(g)<\deg(h)$, all roots of $h$ are $p$-th roots of unity of multiplicity at most $d+1$, and there is a root of multiplicity equal to $d+1$ (all of this assuming that $g/h$ has been reduced to lowest terms.

thm (Beck-Robins ex. 3.25)

Suppose $P$ is a rational $d$-polytope with denominator $p$. Then $$ \operatorname{Ehr}_{P}(z) = \frac{f(z)}{(1-z^p)^{d+1}} $$ where $f$ is a polynomial with nonnegative integral coefficients.

prop (?)

Let $f$ be a quasipolynomial of degree $d$. If $f(n+1)\geq f(n)$ for all $n\in \mathbb{N}$, then the top degree coefficient of $f$ must be constant.