Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Acta Arithmetica / All issues

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.