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Divisor divisibility sequences on tori

Volume 177 / 2017

Joseph H. Silverman Acta Arithmetica 177 (2017), 315-345 MSC: Primary 11B39; Secondary 14G25, 14L99. DOI: 10.4064/aa8381-11-2016 Published online: 22 February 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.

Authors

  • Joseph H. SilvermanMathematics Department
    Box 1917
    Brown University
    Providence, RI 02912, U.S.A.
    e-mail

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