3 added 43 characters in body

I feel like there has to be an easier proof of this, but I just posted a note on my webpage proving the following Theorem. The key is a paper of Sam Payne's.

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 $\prod_i \Phi_{d_i}\left( t_1^{e_1t_1^{e^i_1} \cdots t_n^{e_nt_n^{e^i_n} \right)$ where $\Phi_d$ is the $d$-th cylotomic polynomial and $(e_1, e_2(e^i_1, e^i_2, \ldots, e_ne^i_n) \in \mathbb{Z}_{\geq 0}^n$, with not all the at least one component of $e_i=0$.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. 2 added 33 characters in body I feel like there has to be an easier proof of this, but I just posted a note on my webpage proving the following Theorem. The key is a paper of Sam Payne's. 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$\prod_i \Phi_{d_i}\left( t_1^{e_1} \cdots t_n^{e_n} \right)$where$\Phi_d$is the$d$-th cylotomic polynomial and$(e_1, e_2, \ldots, e_n) \in \mathbb{Z}_{\geq 0}^n$, with not all the$e_i=0$. (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$\mathbb{Q}$, \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.

1

I feel like there has to be an easier proof of this, but I just posted a note on my webpage proving the following Theorem. The key is a paper of Sam Payne's.

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 $\prod_i \Phi_{d_i}\left( t_1^{e_1} \cdots t_n^{e_n} \right)$ where $\Phi_d$ is the $d$-th cylotomic polynomial and $(e_1, e_2, \ldots, e_n) \in \mathbb{Z}_{\geq 0}^n$, with not all the $e_i=0$.

(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 $\mathbb{Q}$, 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.