Algebra Seminar talk
Hilbert’s tenth problem over rings of algebraic integers
In 1900 David Hilbert held his famous lecture entitled „Mathematische Probleme“ before the Second International Congress of Mathematicians in Paris. Among the 23 mathematical problems that he posed during the lecture the tenth problem and its generalizations are the subject of this talk. It states
Given a Diophantine equation with any number of unkown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers.
It took nearly 40 years to formalize the notion of ‘finitary process’ to what today is known as ‘algorithm’. That such an algorithm cannot exist was famously proven by Davis, Putnam, Robinson and Matiyasevič in 1970.
In this talk I will however concern myself with the case of Diophantine equations with coefficients over the algebraic integers and their solvability in algebraic integers. Before formally defining the notion of such an decidability problem, I will explore the vast variety of closely related problems. We will see that the problem becomes easier to decide in ‘algebraically larger’ rings.
Eventhough, H10 still lacks a full dichotomy over all rings of algebraic integers, some structural methods have sucessfully been applied to deduce the undecidablitiy of H10 over some rings from the undecidablity of the problem over the rational integers. These so called vertical methods are based on the techniques developed by Matiyasevič in 1970 and were first presented by Denef and Lipshitz.
If time permits, I will sketch how Denef found a Diophantine relation describing the rational integers over the ring of algebraic integers of a totally real number field.
Slides and source code can be accessed at http://tim6her.github.io/H10-over-rings-of-integers-slides