On the possible ranks of universal quadratic forms over totally real number fields

Abstract: A quadratic form $Q(X_1,...,X_n)$ over the integer numbers is said to be universal if it represents all positive integers, that is, for every $a \in \mathbb{Z}>0$ there is a vector $(\alpha_1,...,\alpha_n)\in \mathbb{Z}^n$ such that $Q(\alpha_1,...,\alpha_n) = a$. The topic of universal quadratic forms is quite classical in arithmetic; for instance, Lagrange's 1770 four square theorem asserts that the sum of four squares is a universal quadratic form over $\mathbb{Z}$. In this talk, we consider the suitable generalization of universality for quadratic forms over the number ring $\mathcal{O}_K$ of a totally real number field $K$ and view some recent results on the possible ranks (number of variables $X_i$) of universal quadratic forms over different families of totally real number fields.

Mathematics

Audience: advanced learners

Barcelona Mathematics Informal Seminar (SIMBa)

Series comments: SIMBa is a youth mathematics seminar organized by graduate students of the Barcelona area. It is aimed towards graduate and last course undergraduate students. Our goals are divulging the knowledge from different branches of mathematics for those interested and promote networking between the attendants.

This seminar is backed by the Faculty of Mathematics and Computer Science at Universitat de Barcelona, Faculty of Mathematics and Statistics at Universitat Politècnica de Catalunya, the Department of Mathematics from Universitat Autònoma de Barcelona, CRM, IMUB and BGSMath.