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Date: January 27, 2016
Time: 10:00-10:50am
Lecturer(s): Francesco Pappalardi (Universita Roma Tre)
Location: University of Lethbridge, UHall C630
Topic: The Distribution of Multiplicatively Dependent Vectors
Description (in plain text format):
Let n be a positive integer, G be a group and let nu = (nu_1,...,nu_n) be an n-tuple of elements of G. We say that nu is a multiplicatively dependent n-tuple if there is a non-zero k-tuple (k_1,...,k_n) of integers for which the product nu_1^{k_1}...nu_n^{k_n} is equal to 1.
Given a finite extension K of the rational numbers, we denote by M(n,K,H) the number of multiplicatively dependent n-tuples of algebraic integers of K* 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* 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 tends to infinity.
For each nu in (K*)^n we define m, the multiplicative rank of nu, in the following way. If nu has a coordinate which is a root of unity we put m=1. Otherwise let m be the largest integer with 2 < m < n+2 for which every set of m-1 of the coordinates of nu 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*, respectively algebraic numbers from K*, of naive height at most H and will consider similar questions for them.
Other Information: Location: C630 University Hall
Contact:
Barb Hodgson | hodgsonb@uleth.ca | (403) 329-2470 | uleth.ca/artsci/math-computer-science
