Divisor Divisibility Sequences on Tori

Abstract

We define the $\textit{Divisor Divisibility Sequence}$ associated to a Laurent polynomial $f\in\mathbb{Z}[X_1^{\pm1},\ldots,X_N^{\pm1}]$ to be the sequence $W_n(f)=\prod f(\zeta_1,\ldots,\zeta_N)$, where $\zeta_1,\ldots,\zeta_N$ range over all $n$'th roots of unity with $f(\zeta_1,\ldots,\zeta_N)\ne0$. More generally, we define $W_\Lambda(f)$ analogously for any finite subgroup $\Lambda\subset(\mathbb C^*)^N$. We investigate divisibility, factorization, and growth properties of $W_\Lambda(f)$ as a function of $\Lambda$. In particular, we give the complete factorization of $W_\Lambda(f)$ when $f$ has generic coefficients, and we prove an analytic estimate showing that the rank-of-apparition sets for $W_\Lambda(f)$ are not too large.