Lethbridge Number Theory and Combinatorics Seminar: Francesco Pappalardi

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Let n be a positive integer, G be a group and let ν = (ν₁,...,ν_n) be in Gⁿ. We say that ν is a multiplicatively dependent n-tuple if there is a non-zero vector (k₁, . . . , k_n) in Zⁿ for which ν₁^k₁ ... ν_n^k_n =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^∗ 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 → ∞.

 

For each ν in (K^∗)ⁿ we define m, the multiplicative rank of ν, in the following way. If ν has a coordinate which is a root of unity we put m = 1. Otherwise let m be the largest integer with 2 ≤ m ≤ n + 1 for which every set of m − 1 of the coordinates of ν 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.