Arity of universal quadratic forms over real quadratic fields

Valentin Blomer, Vítězslav Kala

Published 2017-05-10Version 1

We study the minimal number of variables required by a totally positive definite diagonal universal quadratic form over a real quadratic field $\mathbb Q(\sqrt D)$ and obtain lower and upper bounds for it in terms of certain sums of coefficients of the associated continued fraction. We also estimate such sums in terms of $D$.