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.