# SIMBa: On the possible ranks of universal quadratic forms over totally realnumber fields

El proper dimecres, 6 d’octubre, se celebrarà una nova xerrada —en format virtual— del Seminari Informal de Matemàtiques de Barcelona (SIMBa).

Speaker: Daniel Gil Muñoz
Universitat: Charles University in Prague

Data: Wednesday, October 6th, 2021.
Hora: 12:00,virtual coffee break; 12:20, talk.
Lloc: Zoom (the link will be posted on our website).
Idioma: English.

Títol: On the possible ranks of universal quadratic forms over totally realnumber fields
Resum: A quadratic form $Q\left(X_{1}, \ldots, X_{n}\right)$ 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 $\left(\alpha_{1}, \ldots, \alpha_{n}\right) \in \mathbb{Z}^{n}$ such that $Q\left(\alpha_{1}, \ldots, \alpha_{n}\right)=a$. The topic of universal quadratic forms is quite classical in arithmetic; for instance, Langrange’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.

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