No proper generalized quadratic forms are universal over quadratic fields

Ondřej Chwiedziuk, Matěj Doležálek, Simona Hlavinková, Emma Pěchoučková, Zdeněk Pezlar, Om Prakash, Anna Růžičková, Mikuláš Zindulka

Published 2024-09-12Version 1

We consider generalized quadratic forms over real quadratic number fields and prove, under a natural positive-definiteness condition, that a generalized quadratic form can only be universal if it contains a quadratic subform that is universal. We also construct an example illustrating that the positive-definiteness condition is necessary.