University of Sydney Algebra Seminar

Alina Ostafe (University of New South Wales)

Friday 2 June, 12-1pm, Place: Carslaw 375

Multiplicatively dependent points on curves and applications to algebraic dynamical systems

Bombieri, Masser and Zannier (1999) proved that the intersection of a curve defined over a number field with the union of all proper algebraic subgroups of the multiplicative group \(\mathbb{G}_m^n\) is a set of bounded height (unless this is false for an obvious reason). It is important to note that this set is still infinite as the degree of the points is not bounded.

In this talk we present recent results on multiplicative relations of points on algebraic curves, when restricted to certain proper subfields of the algebraic closure of \(\mathbb{Q}\), complementing those of Bombieri, Masser and Zannier (1999). Some of our initial motivation comes from studying multiplicative relations in orbits of algebraic dynamical systems, for which we present several results.

Furthermore, combined with Hilbert’s Nullstellensatz such results give information about multiplicative relations in reductions modulo primes. We conclude the talk with discussing intersections of orbits with algebraic varieties in reduction modulo primes, and outline some open questions in this direction.