Lifting problem for universal quadratic forms

By V. Kala, Pavlo Yatsyna. Advances in Mathematics, 2018

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We study totally real number fields that admit a universal quadratic form whose coefficients are rational integers. We show that $\mathbb Q(\sqrt 5)$ is the only such real quadratic field, and that among fields of degrees 3, 4, 5, and 7 which have principal codifferent ideal, the only one is $\mathbb Q(\zeta_7+\zeta_7^{-1})$, over which the form $x^2+y^2+z^2+w^2+xy+xz+xw$ is universal. Moreover, we prove an upper bound for Pythagoras numbers of orders in number fields that depends only on the degree of the number field.

Subfields: Algebra and Number Theory, Artificial Intelligence, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b1a2af9d8cd810f6120309e34b8ed5cb5c895761