Divisor Divisibility Sequences on Tori

TL;DR

This paper introduces divisor divisibility sequences derived from Laurent polynomials evaluated at roots of unity, exploring their divisibility, factorization, and growth, with explicit factorizations and bounds on special sets.

Contribution

It provides the first complete factorization of these sequences for generic polynomials and analyzes the size of rank-of-apparition sets.

Findings

Complete factorization for generic coefficients

Bounds on the size of rank-of-apparition sets

Analytic estimates on growth and divisibility properties

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.