Divisor divisibility sequences on tori

Joseph H. Silverman¹

¹ Mathematics Department Box 1917 Brown University Providence, RI 02912, U.S.A.

Acta Arithmetica, Tome 177 (2017) no. 4, pp. 315-345

Résumé

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.

Détail

DOI : 10.4064/aa8381-11-2016

Keywords: define divisor divisibility sequence associated laurent polynomial mathbb ldots sequence prod zeta ldots zeta where zeta ldots zeta range nth roots unity zeta ldots zeta generally define varlambda analogously finite subgroup varlambda subset mathbb * investigate divisibility factorization growth properties varlambda function varlambda particular complete factorization varlambda has generic coefficients prove analytic estimate showing rank of apparition sets varlambda too large