Quadratic Diophantine equations (sums of integer squares ) have fascinated mathematicians for centuries however even today some the finer local to global questions are not understood, and the complexity of finding solutions is challenging. We will explain and review these features and highlight some applications, for example to quantum computation with the construction of optimal universal quantum gates.
A cubic polynomial equation in four or more variables tends to have many integer solutions, while one in two variables has a limited number of such solutions. There is a body of work establishing results along these lines. On the other hand very little is known in the critical case of three variables. For special such cubics, which we call Markoff surfaces, a theory can be developed. We will review some of the tools used to deal with these and related problems when the integers are replaced by integers in say a real quadratic field.
The intersection of the division group of a finitely generated subgroup of a torus with an algebraic subvariety has been understood for some time (Lang, Laurent,…). After a brief review of some of the tools in the analysis and their recent extensions (Andre’-Oort Conjectures ), we give some old and new applications; periodicity of Betti numbers, algebraicity of Painleve’ equations, and the additive structure of spectra of quantum graphs.