Pacific Institute for the Mathematical Sciences - PIMS

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Let n be a positive integer, G be a group and let ν = (ν1,...,νn) be in Gn. We say that ν is a multiplicatively dependent n-tuple if there is a non-zero vector (k1, . . . , kn) in Zn for which ν1k1 ... νnkn =1.

Given a finite extension K of Q, we denote by Mn,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 Mn,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 Mn,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.

Location: C630 University Hall