Searching for Rational Points on Varieties over Global Fields

The celebrated Hilbert\u27s 10th problem asks for an algorithm to decide whether a system of polynomial equations with integer coefficients has a nontrivial solution. A famous theorem of Y. Matiyasevich (1970) states that no such algorithm exists. Moreover, a result of J. P. Jones (1980) implies that no algorithm exists even for a single homogeneous polynomial of degree four with rational coefficients. On the other hand, such algorithms have been proven to exist for systems of linear equations, as well as for a single quadratic equation. We will discuss a certain approach to the problem of searching for rational solutions of linear and quadratic equations, which also leads to an investigation of rational points on some related varieties over number fields and function fields. This approach involves height functions, which are common tools of modern arithmetic geometry