Hensel minimality

Studying rational points on algebraic varieties is one of the most fundamental topics in algebraic geometry and number theory. The proposed project fits into this research topic. For a rational number q=a/b with a and b coprime, we define the height of q to be the maximum of |a| and |b|. In some sense, this measures the complexity of the number q. For a tuple (q_1, ..., q_n) of rational numbers, the height is defined to be the maximum of the heights of the q_i. For a polynomial equation f(x_1, ..., x_n) = 0 in several variables, we are interested in upper bounds for the number of solutions of height at most B, say. This line of research was initiated by Bombieri and Pila and further developed by Heath-Brown, Salberger and Walsh. It has since found many applications within algebraic geometry and number theory.

Instead of working with rational numbers, one could also work with rational function fields in one variable over a finite field, F_q(t), and ask the same type of question. There are many analogies between these two fields, and theorems in one setting often have a counterpart in the other. Therefore, it seems believable that these results on upper bounds on points of bounded height in characteristic 0 can be extended to positive characteristic as well. The project will study these questions and applications to cryptography in more detail.