## Divisor divisibility sequences on tori

### Volume 177 / 2017

#### Abstract

We define the *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
$W_\varLambda(f)$ analogously for any finite subgroup
$\varLambda\subset(\mathbb{C}^*)^N$. We investigate divisibility,
factorization, and growth properties of $W_\varLambda(f)$ as a function
of $\varLambda$. In particular, we give the complete factorization of
$W_\varLambda(f)$ when $f$ has generic coefficients, and we prove an
analytic estimate showing that the rank-of-apparition sets for
$W_\varLambda(f)$ are not too large.