Let $P$ be a subset of $\mathbb N^d$ (or of some normal pointed affine semigroup), and suppose that $f:=\sum\_{p\in P}\ t^p\in\mathbb Z[[t\_1,\ldots,t\_d]]$ is a rational function. What can be said about the structure of $P$? In particular, must $P$ be a finite (disjoint) union of finitely generated modules over affine sub-semigroups?  

Equivalently, must $P$ be a finite (disjoint) union of intersections of a rational polyhedron in $\mathbb N^d$ with a sublattice of $\mathbb Z^d$?


(My proximal motivation for asking is the appearance of both conditions in [Guo & Miller, *Lattice point methods for combinatorial games*](http://www.math.duke.edu/~ezra/Games/cgt.pdf). One would like to speak on the level of rational generating functions, but the tools they develop only let them get at sets with a given decomposition into modules for sub-semigroups.)



Which sets of lattice points have rational generating functions?
I feel like there has to be an easier proof of this, but I just posted [a note](http://www.math.lsa.umich.edu/~speyer/PowSerNote.pdf) on my webpage proving the following Theorem. The key is a [paper of Sam Payne's](http://arxiv.org/abs/0706.2438).


Let $f(t\_1, \ldots, t\_n)/g(t\_1, \ldots, t\_n) = \sum a(d\_1, \ldots, d\_n) t\_1^{d\_1} \cdots t\_n^{d\_n}$ be a rational function with coefficients in $\mathbb{Q}$. Let $\mathbb{C}\_p$ be the completion of the algebraic closure of $\mathbb{Q}\_p$, so $\mathbb{C}\_{\infty}$ means the standard complex numbers.
We define a function $\phi: \mathbb{Z}\_{\geq 0}^n \to \mathbb{Q}$ to be a quasi-polynomial if $\mathbb{Z}\_{\geq 0}^n$ can be partitioned into finitely many sets $S\_k$, each one the translate 
of a finitely generated semi-group, such that the restriction of $\phi$ to each $S\_k$ is a polynomial.


**Theorem:** The following are equivalent:


(1) The polynomial $g$ factors as $\prod\_i \Phi\_{d\_i}\left( t\_1^{e^i\_1} \cdots t\_n^{e^i\_n} \right)$ where $\Phi\_d$ is the $d$-th cylotomic polynomial and $(e^i\_1, e^i\_2, \ldots, e^i\_n) \in \mathbb{Z}\_{\geq 0}^n$, with at least one component of $e^i$ nonzero for each $i$.


(2) The function $(d\_1, \ldots, d\_n) \mapsto a(d\_1, \ldots, d\_n)$ is a quasi-polynomial.


(3) There are constants $C$ and $D$ such that 
$$|a(d\_1, \ldots, d\_n)|\_{\infty} \leq C \left( \sum d\_i \right)^D$$
and, for every finite prime $p$, there is a constant $C\_p$ such that
$$|a(d\_1, \ldots, d\_n)|\_{p} \leq C\_p.$$


(4) For every absolute value $| \ |\_p$ on $\mathbb{Q}$ (including the archimedean norm), there are no zeroes of $g(t\_1, \ldots, t\_n)$ in the open polydisc $\{ (u\_1, \ldots, u\_n) \in \mathbb{C}\_p : |u\_1|, |u\_2|, \ldots, |u\_n| < 1 \}$.


In your setting, suppose that $\sum\_{d \in P} t\_1^{d\_1} \cdots t\_n^{d\_n}$ is rational. Let $\chi\_P$ be the characteristic function of $P$. It clearly obeys condition (3). So the theorem states that $\chi\_P$ is a quasi-polynomial. Each of the polynomials making it up must have degree $0$, as it only assumes two values. So the support of $\chi\_P$ (that is to say, the set $P$) must be a union of translates of finitely generated semi-groups.


Can someone tell me whether this is new? I think it might be worth publishing, if so.


