Representing rational integers by generalized quadratic forms over quadratic fields
Number Theory
2026-03-24 v2
Abstract
We investigate generalized quadratic forms with values in the set of rational integers over quadratic fields. We characterize the real quadratic fields which admit a positive definite binary generalized form of this type representing every positive integer. We also show that there are only finitely many such fields where a ternary generalized form with these properties exists.
Keywords
Cite
@article{arxiv.2403.07171,
title = {Representing rational integers by generalized quadratic forms over quadratic fields},
author = {Ondřej Chwiedziuk and Matěj Doležálek and Emma Pěchoučková and Zdeněk Pezlar and Om Prakash and Giuliano Romeo and Anna Růžičková and Mikuláš Zindulka},
journal= {arXiv preprint arXiv:2403.07171},
year = {2026}
}
Comments
16 pages + 11 pages appendix; moved technical details to the appendix