Time: 14:15 on 2 Mar, 2017
Place: Speigelgasse 5, 05.002
Igor Shparlinksi (Univ. of New South Wales/MPI-Bonn)
Multiplicatively dependent algebraic numbers
(joint work with Alina Ostafe, Francesco Pappalardi, Min Sha, Cam Stewart) We discuss various questions related to the distribution of vectors of algebraic numbers (u_1, …., u_n) which are multiplicatively dependent. In particular, we present some counting results for the number of such vectors of degree d and height h (from a fixed number field K and from Q-bar). We also give both sided estimates on their density in R^n.Finally we give an analogue of a result of Bombieri-Masser-Zannier (1999) proving the boundedness of the “house" of the shifts v from the abelian closure of a given number field K, for which the vector (u_1-v, …., u_n-v) becomes multiplicatively dependent,rather than the boundedness of their height as in BMZ’99 (but for shifts v from Q-bar). This result has applications to multiplicative dependence in orbits of polynomial dynamical systems, generalising those on roots of unity.