School of Mathematics & Physics, University of Lincoln
On Tuesday the 10th of December 2019, Andre Macedo (University of Reading) gave a research seminar as part of the Advanced Topics of Mathematics and Mathematics Seminar. His talk details were as follows:
Title: Local-to-global principles for solving Diophantine equations
Abstract: The modern approach to the question of whether a polynomial equation admits rational solutions is to first check whether local solutions exist at every completion of the rationals (a finite computation), and then check whether the Hasse principle holds. If the Hasse principle holds, then the existence of local solutions everywhere guarantees the existence of a rational solution.
In this talk, I will introduce the Hasse principle and give examples of some nice families of equations for which the principle holds. I will then discuss this local-to-global approach for a natural class of equations coming from norms of number fields (the basic objects of algebraic number theory). I will provide an overview of what is known on this topic, the main techniques one uses to study it and present some recent developments and lines of research.