# Lethbridge Number Theory and Combinatorics Seminar: Francesco Pappalardi

## Topic

The Distribution of Multiplicatively Dependent Vectors

## Details

(Please note that this abstract has been reformatted for the web, and some of the original mathematical typesetting may have been lost. Click here [.pdf] for original formatting.)

Let n be a positive integer, G be a group and let Î½ = (Î½

_{1},...,Î½

_{n}) be in G

^{n}. We say that Î½ is a

*multiplicatively dependent n-tuple*if there is a non-zero vector (k

_{1}, . . . , k

_{n}) in Z

^{n}for which Î½

_{1}

^{k1}... Î½

_{n}

^{kn}=1.

Given a finite extension K of Q, we denote by M

_{n,K}(H) the number of multiplicatively dependent n-tuples of algebraic integers of K

^{âˆ—}of naive height at most H and we denote by M

^{*}

_{n,K}(H) the number of multiplicatively dependent n-tuples of algebraic numbers of K

^{âˆ—}of height at most H. In this seminar we discuss several estimates and asymptotic formulas for M

_{n,K}(H) and for M

^{*}

_{n,K}(H) as H â†’ âˆž.

For each Î½ in (K

^{âˆ—})

^{n}we define m,

*the multiplicative rank of Î½*, in the following way. If Î½ has a coordinate which is a root of unity we put m = 1. Otherwise let m be the largest integer with 2 â‰¤ m â‰¤ n + 1 for which every set of m âˆ’ 1 of the coordinates of Î½ is a multiplicatively independent set.

We also consider the sets M

_{n,K,m}(H) and M

^{*}

_{n,K,m}(H) defined as the number of multiplicatively dependent n-tuples of multiplicative rank m whose coordinates are algebraic integers from K

^{âˆ—}, respectively algebraic numbers from K

^{âˆ—}, of naive height at most H and will consider similar questions for them.

## Additional Information

Location: C630 University Hall

Francesco Pappalardi (UniversitÃ degli Studi Roma Tre)

This is a Past Event

Event Type

**Scientific, Seminar**

Date

**January 27, 2016**

Time

**-**

Location

University of Lethbridge