## 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 Hilbertâ€™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.